Castling is a new move in chess. Well, relatively new. It became popular in its current form around the 1700s to 1800s. The basic idea was to speed up the game. Because the King is a slow moving piece, players in the past would have to spend 3 or 4 moves setting up the castled position. Instead of doing all of this, now a days, players can castle in one move. Players can castle either towards the right side of the board, called the Kingside, or to the left side of the board, called the Queenside.
As a brief aside:
From the d file to the left is called the Queen side. From the e file to the right is called the King side. These names arent official and have no real impact on the game. It is simply a way for chess players to talk or write about various chess concepts in a simple way. For example, "I was attacking on the Queen side and he was counter attacking on the King side," or "I had a Queen side pawn majority and he had a King side pawn majority."
Back to castling.
Before a player can castle they must make sure there are no pieces in the way.
In this position, note that the squares f1 and g1 are clear. White's Knight and Bishop have moved off of these squares, and now White can castle King side.
When you castle Kingside, you place your King on g1. Then you place your Rook on f1.
There you have it, quick learning beginner. You now know how to castle King side!
Queen side castling is also possible. It is very similar to King side castling.
Notice how in this position the squares b1, c1, and d1 are all clear. With nothing blocking the way, White can castle Queen side. To castle Queen side, White places his King on c1, and his Rook on d1.
Simple!
Well, unfortunately, not quite that simple. There are a few other rules of castling.
The first rule is that neither the King nor the Rook the player wants to castle with can move. If either piece moves, castling is not possible.
Looking at the above example of Queen side castling, if White move his Rook to b1, and then back to a1, he will not be allowed to castle Queen side. However, if he moves his Light Squared Bishop off of the f1 square, he will still be able to castle King side.
If he moves his King to d1, though, he will not be allowed to castle either King side or Queen side.
Finally, the King is not allowed to move through check to castle. Check, and checkmate were explained in A Beginner's Guide, Part 3, Winning. Keep in mind that the Rook is allowed to move through an attack, though.
As a beginner, dont worry too much about moving through Check. Just know that if the computer is not allowing you to castle, this may be why.
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