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According to the rules of chess, one type of position declared to be a draw is stalemate. Here are some others.
Perpetual check
A player hopelessly behind in material can sometimes save the game by checking on every move, where the opponent has no way to stop the checks. In the diagram, Black to move would win easily, but White to move checks with the Queen on c6, d5, or e4. Black can only intersect the line of the attack by playing Qb7.
White then checks safely on the eighth rank, after which Black can only play Qb8 -- other moves would lose the Queen. White checks again on the same square used the first time -- c6, d5, or e4 -- and declares a draw by perpetual check.
The example in the diagram is from Howard Staunton's Chess-Player's Handbook, written in the 19th century..
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Triple repetition
When a position is repeated three times with the same player to move, the player on the move can declare a draw. In the diagram Black is in check and must move. Any move other than Kf8 lets the white King escape from the corner, followed shortly by h8Q and a quick checkmate.
After 1... Kf8, White can play 2.Ne6+, but Black continues with 2...Kf7, keeping the black King in the corner. The players continue 3.Nd8+ Kf8 4.Ne6+ Kf7 5.Nd8+, when Black can claim a draw by triple repetition. Black is to move and 5... Kf8 will repeat the position for the third time.
This example may look like perpetual check, but there's a big difference. Given the opportunity, White has enough material to win here. Even if the Knight moves away to avoid giving check by Nb7, the black King continues to shuttle between f7 and f8. White can never lose the tempo needed to force the black King away from g7/g8. White can avoid repeating all positions three times, but the '50 move rule' (see below) will eventually allow Black to claim the draw.
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Insufficient mating material
It should be clear that King against King, with no other pieces, is a draw. Neither side can check or give checkmate.
The diagram is King and Knight against a lone King. To announce checkmate, White must give check plus control the three squares around the cornered black King. The white King can control only two squares, while the Knight is incapable of checking and controlling the last square at the same time.
The same logic applies to King and Bishop against a lone King. The game is a draw.
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50 move rule
If a game continues 50 moves without a Pawn move or a capture, the game is drawn. That means 50 moves for White and 50 moves for Black.
The diagram shows an endgame of Rook and Bishop vs. Rook. This is a theoretical endgame known to be drawn if Black plays correctly. In practice it's a difficult endgame for the defense to conduct and there are many ways to go wrong. If Black can survive for 50 moves, the game is a draw.
Computer analysis of theoretical endgames has discovered positions where the stronger side needs more than 50 moves to win. For those exceptional positions the 50 move rule has been altered to allow sufficient time to win.
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Mutual agreement
Some positions offer absolutely no winning chances to either player. It would take some very bad moves for either player to lose from the position in the diagram. The game could be drawn by mutual agreement. This is, in fact, how most games are drawn.
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