Basics

If you are new to Chess960, you should first read Erik's inaugural article:
http://www.chess.com/forum/view/chess960-chess-variants/chess960-explained , and follow the Wiki link at the bottom of that article for a complete set of rules.
 

Rules of Castling in 960

http://en.wikipedia.org/wiki/Chess960#Rules_for_castling

Just drag your king and drop it over the final king-position (c1/c8 or g1/g8) - there should not be any piece in-between the king and the rook (and obviously the king should not be in check). In case the original king-position and the final king-position are adjacent, the software gets confused whether you want simply to move the king or castle. In those cases, try dropping the king over the rook.

Past Basics
If you already know the rules, and keen to learn more, there is no article better than that of IM David Pruess:
http://www.chess.com/article/view/chess960-the-opening-makes-a-comeback

FEN string in Chess960
The first part or string (as in standard) is the position of the pieces (upper case for white, lower case for black, empty spaces by a numeric denoting number of adjoining empty spaces, each rank separate by a forward slash "/"). An example of first part is "brkqnrnb/pppppppp/8/8/8/8/PPPPPPPP/BRKQNRNB".

The second (character after a space) is whose turn it is to move ("w" if it is white's turn, "b" if it is black's turn; it is always either w or b).

The third is castling availability, and is different from that of standard chess. Currently chess.com editor malfunctions if you put anything other than "-" here! It is a string 1-4 characters, upper case for white, lower case for black. Whereas in standard chess "K" denotes white can castle kingside, "q" denotes black can castle queenside, in 960 these are the notations for squares on which the rooks sit. Thus "F" denotes white can castle with rook sitting on f1; "fc" denotes black can castle kingside with rook sitting on f8, queenside with rook sitting on c8; "GA" means white can castle kingside with rook sitting on g1, queenside with rook sitting on a1; "GAa" means white can castle both sides (as stated before), whereas black can only castle queenside with rook sitting on a8; "-" denotes that neither side can castle; etc. In the opening position, it is always four characters (first two upper case, next two lower case) of the form GAga, FCfc, etc.

The fourth is en passant target square. "-" (usual) means none (no en passant opportunity), else if a pawn just made a two-square move, it is the notation of the square immediately behind it (where the capturing pawn will go). e.g.: If white played b4, and there is a black pawn on a4 or c4, then this will be b3, meaning the black pawn can capture e.p. and go to b3. Again, if black played g5, and there is a white pawn on f5 or h5, then this will be g6, meaning the white pawn can capture and go to g6.

We will ignore the fifth, as it is of no use for all practical purposes (chess.com's gamut of diagram tools also ignore it ) - just consider it 0.

The sixth is the full-move number (simply the move number which we are on), and incremented by one after black's move is done. Thus 8 means currently it is (white's or black's) eighth move.

Bypassing bugs in CHESS.COM's tools for 960

You will often find difficulties in creating/editing chess diagrams - especially with FEN string. One way is to use the entire PGN, then "focussing" (highlighting) the current move. Another way is to copy the PGN to notepad, make the necessary corrections, and to (re)create the diagram from the modified PGN.

Opening Principles

Which do you think is the hardest part of Chess960? The opening, of course! So if your appetite is whetted by David's article, here is some more points:

(1) Continuing in the vein of David's article, find the undefended/underdefended pawns (if any) on both sides. They are your object of attack and defense in the first few moves.

(2) Find the minimum number of single-square (one-rank) pawn moves that can open the diagonals for your bishops and queen.

(3) Want to control the center? Consider pushing 2 squares the pawns in front of rooks and queen - see if pushing them does not create any disadvantage in the neighborhood or elsewhere.

(4) Find the best squares for your knight (rule of thumb - toward the center is better). Sometimes you might have to move a knight more than once to reach a desired vantage point - try to defer that second move till most of your minor pieces are developed.

(5) Find the most economical way to develop your queen.

(6) Consider whether castling is needed at all - if so, to which side?

(7) What are the weak spots of your and your opponent's position if you/he/she castles, and if you/he/she does not castle?

(8) Think of attack only after fully developing all pieces. The exception is in situations where the opponent is grossly backward in development, and wasting moves will alow him/her to catch up.

Here is an interesting (but not necessarily good) link on more about openings in 960:

http://chess960jungle.blogspot.de/2011/05/chess960-naming-pawn-moves.html

(Adding illustrative diagrams is too much of a hasstle, and will be added only if requested - no timeline).

Watch for knight opposition. In the standard opening set up, you king side knight covers the same central squares as your opponent's queen side knight when they are both developed. This is not always the case in Chess960, there is the potential for the knights to make an early attack unopposed.

If you are having trouble loading a Chess.com Chess960 pgn into another program, check the FEN string. Chess960 pgns generated by Chess.com use KQkq for the castling indicator. Change KQ to the rook squares as specified in the first post.

@laskerfan that was a long comment