See The Oxford Companion to Chess (on reserve for FS23j) for plenty of further definitions and great examples.

Directmate (n.): A chess problem with the stipulation of the form ``Mate in N'' for some N=1,2,3,...; that is, ``White to play and mate in N moves''. This means that White must demonstrate checkmate in at most N moves against any Black defense. Usually N>1.

Fairy problem (n.): Any chess problem with a stipulation other than Mate, Helpmate, or Selfmate, or using unorthodox chessmen (such as the Grasshopper and non-(2,1) leapers), rules (as in series-movers), or boards.
Retrograde problems are usually classified as fairy, even if presented with an orthodox stipulation. A prototypical example of this is the position White Ke1,Qg3,Ra1,Pd2f2g2, Black Kg1: Mate in One. The solution Ke2# is unique, because 0-0-0# is demonstrably illegal -- the White King must have moved to let the Black King reach g1 via f1.

Grasshopper (n.): An unorthodox piece, extensively used in fairy problems since its invention by T.Dawson in 1912. Here is the Oxford Companion's definition of the Grasshopper's move: ``It may be moved any distance along ranks, files, and diagonals to occupy, or capture on, a square immediately beyond an intervening man of either colour; it can neither be moved unless it hops[,] nor hop over two men.'' For example (Umnov, Feenschach 1979): White Kd8; Black Kd5, Grasshopper a1: Help-stalemate (q.v.) in 5 by 1 Kc6! Ke7! 2 Kb7 Kf6! 3 Gg7 Ke6! 4 Ga7 Kd7 5 Ka8 Kc8 stalemate! Note that White avoided 3...Ke7?, which would prevent 4 Ga7, and 5...Kc7?, which would not be stalemate (6 Gd7). In chess diagrams the Grasshopper is usually represented by an upside-down Queen figurine.
Conventionally, when there is a Grasshopper in the diagram, pawns may promote to Grasshoppers (besides the usual Q/R/B/N); likewise for other unorthodox pieces.

Helpmate (n.): A cooperative problem genre, in which Black and White cooperate to get Black checkmated in the set number of moves. Usually Black moves first. See UNDERPROMOTION below for an example. An even simpler one is the position White Kc8,Pb6; Black Ka8,Pa7 [A.H.Kniest, Deutsche Märchenschach-Zeitung, 1932], a Helpmate in Two with the unique solution 1 a6 b7+ 2 Ka7 b8Q#. (Note the move numbering, which reverses White's and Black's usual roles.) If White is to move first, the stipulated number of moves would be a half-integer; for an (extreme) example, a Mate in One is equivalent to a Helpmate in 0.5. The fact that the above position also has a solution as a Helpmate in 0.5 (which would be notated 1...b7#) does not render it invalid -- remember that the WTM and BTM versions of a given diagram are different positions! -- but if there were a shorter solution with Black moving first then the problem would be deemed unsound.
The Oxford Companion attributes the invention of the Helpmate to Max Lange in 1854. All moves in a helpmate must be legal: the ``cooperation'' does not extend to putting one's own King in check (trusting that it not be captured) or playing other kinds of illegal moves such as capturing one's own pieces (trusting that the other player will not object).

Help-stalemate (n.): a problem genre analogous to the helpmate except that White and Black cooperate to get Black stalemated rather than checkmated. Help-stalemates follow the same conventions as helpmates regarding stipulation, move numbering, and legality.

Ideal mate (n.): a pure mate (q.v.) using all the men on the board.

Key, keymove (n.): the first move of the solution of a problem or study.

Leaper (n.): a generalized Knight. The Knight is a (1,2) leaper. In medieval chess, (1,1) and (2,2) leapers appeared where we now have Queens and Bishops. [See HISTORY OF CHESS in The Oxford Companion to Chess. The (1,1) and (2,2) leapers were called ``fers'' and ``alfil'' respectively. Some languages retain versions of these names for the modern Queen and Bishop. For instance, the Italian for Bishop is ``alfiere''; the Russian is ``slon'' -- which means ``elephant'', same as the Arabic ``al-fil''. For consistency we call the (1,1) a leaper even though its move is too short to actually leap over anything; likewise the (1,0).] Other leapers are used on occasion in problems, sometimes with traditional names such as Wazir (1,0), Camel (1,3), Zebra (2,3), Giraffe  (1,4). See for instance this site. How many possible leapers are there on the 8-by-8 chessboard, or more generally on the n-by-n? A ``five-leaper'' combines the powers of a (5,0) and (3,4) leaper -- remember Pythagoras! What is the one other such combination possible on the 8-by-8 chessboard?

Legal position (n.): a position that can be reached from the initial array by game consisting entirely of legal moves, however bizarre. Conventionally every chess problem should have a legal position. Naturally then, an illegal position is a position that cannot be reached by a legal game. For instance, a position in which one side has more than 8 pawns, or has both White and Black Kings in check, is illegal (why?). So is any position with a White Bishop on a1 and White pawn on b2 (why?), such as the following mutual Zugzwang (q.v.), which Lewis Stiller discovered in the course of an exhaustive computer search: White Kg6, Bh1, Pg2; Black Kg4, Pg3. The Kniest position White Kc8,Pb6; Black Ka8,Pa7 (seen above under Helpmate) is legal BTM, but not WTM since Black is in ``retro-stalemate'': Black could not have made a legal move to reach this position. [Thus this position can be set as a Helpmate in Two but not a Mate in One (or ``helpmate in 0.5'').] There are positions that can be recognized as illegal only after extensive retrograde analysis. To prove that a position is legal, one need only exhibit a single legal game reaching the position; such a game is called a proof game. Some retrograde analysis may still be needed to construct a proof game.

Miniature (n.): a problem or study with at most 7 men in the diagram position.

Model mate (n.): a pure mate (q.v.) involving all the attacker's men except possibly the King and pawn(s).

mZZ (abbr.): mutual Zugzwang. See "Zugzwang" below.

Piece (n.): Any of a King, Queen, Rook, Bishop, or Knight -- though when chessplayers talk about ``winning a piece'' or ``losing a piece'' they almost always mean a ``minor piece'', that is, a Bishop or Knight. At any rate, a pawn is not counted as a ``piece''. Grasshoppers, Camels, etc. are usually deemed to be pieces. To include pawns as well as pieces, write ``chessmen'' (or simply ``men'' -- as in ``5-man ending''), or occasionally ``units''.

Ply (n): a.k.a. ``half-move'', that is, a Black or White move. Usually applied in the context of how far one is analyzing the position; for instance, to solve a Mate in 5 requires looking ahead 9 plies, while a Helpmate in 5 requires 10 (not counting the two extra plies for verifying the mate).

Pure mate (n.): a checkmate in which each of the squares in the field of the checkmated King is controlled exactly once. Double checks are allowed if both checking units are needed for checkmate; a square blocked by a friendly piece may not be attacked unless the attack is a pin necessary for the checkmate (which is then a ``pure pin-mate''). For example, the position White Kg6,Ra8,Bd6 vs. Black Kf8 is a pure double-check mate; and adding a Black Bishop on e7 makes it a pure pin-mate, but with a Black Pawn instead of the Bishop the mate would be neither pure nor a pin-mate. A pure mate is a ``model mate'' if it uses all the attacker's men except possibly the King and pawn(s); it is an ``ideal mate'' if every unit on the board is required for the checkmate. Each of the pure mates above is ideal (and thus a fortiori a model mate).

Series-mover (n.): A problem in which one side makes a series of moves, unanswered except (usually) for a single move by the other player at the end of the series. All moves in the series must be legal, and none may be check except the last move. In a series helpmate (or help-stalemate) in N, Black makes N consecutive moves, and White answers with a single move that checkmates (or stalemates) Black. For example, the Kniest position White Kc8,Pb6; Black Ka8,Pa7, introduced above as a Helpmate in Two, is also a Series Helpmate in 8, with the unique solution 1 a5 2 a4 3 a3 4 a2 5 a1B! 6 Be5 7 Bb8 8 Ba7 b7#. [Note the move numbering; such long solutions are often abbreviated, for example 1-5.a1B 6.Be5 7.Bb8 8.Ba7 b7#, or even 1-5.a1B 6-8.Ba7 b7#.] In a series selfmate (or self-stalemate) in N, White makes N consecutive moves, at the end of which Black makes a single move and is forced to checkmate (or stalemate) White with that move. Rarer is a direct series mate (or series stalemate) in N, in which White makes a series of N moves, the last of which checkmates or stalemates Black. Example: White Kg6, Pawns g2,h2; Black Kh8: Series mate in 9, how many solutions?

Selfmate (n.): A problem genre in which White forces an uncooperative Black to checkmate White. (In most positions this is much harder than giving checkmate by force!) In a Selfmate in N, White plays first, forcing Black to give checkmate on Black's N-th move or earlier. A relatively simple example: White Kc1, Qf5, Ne1; Black Kh8, Bd1, Pawns a2,c2,c3,e2. Selfmate in Two by 1 Qf7!, and now 1...a1Q(R) is checkmate, while 1...a1B(N) is answered by 2 Qg6! Bb2(Nb3)#. Not 1 Qg6!? a1N! 2 Qf7 Nb3+ (3 Q:b3).

Self-stalemate (n.): A problem genre analogous to the Selfmate, except that White forces an uncooperative Black to stalemate (rather than checkmate) White. In a Self-stalemate in N, White plays first, forcing Black to give stalemate on Black's N-th move or earlier. Example: White Kh3 Qd5, Black Kg1 Pf6, Selfmate in 13. [Such a long Selfmate can be challenging to solve, let alone compose, by hand; I found it by exhaustive computer search some years ago. Fortunately in this case all Black's moves are forced, so it's easy at least to confirm that the solution works, even if it may be far from obvious that it is unique.] 1 Qf5 Kh1 2 Kg3 Kg1 3 Qf2+ Kh1 4 Qf3+ Kg1 5 Qf5 (completing a triangulation) Kh1 6 Kf2 Kh2 7 Qf3 f5 8 Kf1 f4 9 Qh5+ Kg3 10 Kg1 f3 11 Qg5+ (again exploiting the Pawn to constrict Black's King) Kh3 12 Kh1 f2 13 Qg3+ Kxg3. According to the computer search, 13 is the maximal length of a sound self-stalemate with KQ/KP(f).

Stipulation (n.): The goal of a chess problem or study; for instance, ``[White to play and] Mate in 3 [moves]'', ``White to play and win'', ``Helpmate in 5 [moves]'', or ``series help-stalemate in 11 [moves]''.

Study (n.): A legal chess position with the stipulation that White is to force a win or a draw against best Black play. The solver need only reach a theoretical win or draw; the actual checkmate (or stalemate or perpetual check) need not be demonstrated, and the solution must be unique only until the won or drawn position is reached. The Saavedra position (Kb6,Pc6/Ka1,Rd5) is a study with the stipulation ``White to play and win''. The length of the solution of a study is not stipulated (and need not be the same in different main branches of the analysis); 5-15 moves is typical, but I've seen solutions as short as 2 moves and as long as 50+. Usually, but not always, the position is WTM. Studies are sometimes also known as ``endgames'' (or ``endgame studies''), and are often similar in appearance to positions typical of the final stage of a chess game (to achieve economy of material). But the solutions of studies are more clear-cut and often more tactical than the analysis of endgame positions encountered in competitive play.

Underpromote (v.), underpromotion (n.): promote/promotion to a piece other than a Queen; that is, a Knight, Bishop, or Rook. Some authorities insist that true underpromotion cannot be to a Knight, since the Knight has powers that the Queen lacks. In practical play, and also in directmate problems and endgame studies, Rook or Bishop promotions must be motivated by stalemate. (Remember the Rook underpromotion in the Saavedra study! Can you construct a position in which White must promote to a Rook or Bishop in order to draw?) They are very rare in practical play, and prized in problems and studies. In other problem genres, such as cooperative problems, series problems, and selfmates, there may be other reasons for either side to promote a pawn to Bishop or Rook rather than Queen. For example: White Kh2, Bd1, Pawn h6; Black Kh4, Pawn f4: Helpmate in 3.

WTM (abbr.): White to move.

Zugzwang (n., Ger.: literally, obligation to move): A position in which at least one player would like to pass the move. A Zugzwang is mutual (a.k.a. reciprocal) if it affects both players; that is, if the WTM outcome is strictly worse for White than the BTM outcome. [Some authorities, including the Oxford Companion, use ``Zugzwang'' only for what we call a mutual Zugzwang, and refer to an ordinary Zugzwang as a ``squeeze''.] A mutual Zugzwang is sometimes said to be a ``full-point Zugzwang'' if White wins BTM but loses WTM. This refers to the usual system for scoring a chess game as 1, 0.5, or 0 points for a win, draw, or loss: in most mutual Zugzwangs the obligation to move costs only half a point. The canonical example of a ``full-point Zugzwang'' is a ``trébuchet'' position, such as White: Kf5 and Pawn e4, Black: Kd4 and Pawn e5.

zz or ZZ (abbr.): Zugzwang. Sometimes also mZZ or MZZ, mutual Zugzwang.