UCI (short for Universal Chess Interface) is the standard for graphical chess programs to communicate with chess engines.
PGN (short for Portable Game Notation) is the standard format for recording a game in a text file that is processible by computers.
Example: game.pgn
SAN (short for Short Algebraic Notation) is notation where each move of a piece is indicated by the piece's uppercase letter, plus the coordinates of the destination square.
For example, Be5 (bishop moves to e5), Nf3 (knight moves to f3). For pawn moves, a letter indicating pawn is not used, only the destination square is given. For example, c5 (pawn moves to c5).
LAN (short for Long Algebraic Notation) is fully expanded algebraic notation, both the starting and ending squares are specified. For example: e2e4, e1g1 (castling), e7e8q (promotion) A form of long algebraic notation (without piece names) is also used by the UCI standard.
FEN (short for Forsyth-Edwards Notation) is the standard notation to describe positions of a chess game. FEN differs from the PGN because it denotes only a single position instead of the moves that lead to it.