Not every game ends with a checkmate. Many masters, before they are "mated," choose instead to resign. They believe that there is no longer any hope, so they award the win to their opponent. In tournaments, wins receive one point, recorded 1-0 when white wins and 0-1 when black wins. Many beginners are confused by such resignations. "Why did he resign here?" I often hear from my students. Figuring out the reason can be very instructive. And sometimes, master mistakenly resign when they were not losing. That is quite rare, but it's fun to see and talk about!
Between evenly matched opponents, chess games will often end in draws. So what's a draw? That's when nobody wins. In tournaments, draws are recorded as ½ - ½, essentially half a point rather than the full point for a win. There are six kinds of draws to discuss briefly.
(1) Draws by agreement
This kind of draw is the most common. At any time in the game, you can offer a draw to your opponent. Be mindful of good etiquette, however. Don't offer a draw on every move! That's just flat-out rude. And be sure to offer the draw correctly. Make your move, offer the draw ("I offer a draw" is fine) and only then press your clock (assuming that you are using a chess clock). An offer of a draw without making a move is NOT a legal offer and can be ignored.
Here's a famous grandmaster draw, a game drawn quickly. This game was famous because Tal already had 12 points in their 1960 world championship match. This draw resulted in the crowning of a new world chess champion! Tal had steered the game into a position in which Black has no meaningful chances to win, and Tal also had a large lead in the match.
In chess, each side must move. If the player who must move has no legal move (and is NOT in check), the game ends as a draw, a STALEMATE. Some stalemates are quite common. Here's an example of a king and pawn endgame that ends as a stalemate. At the end of the game, the Black king has no moves and is NOT in check.
Many examples are much more complicated and can be quite exciting.
(3) Perpetual check
Draws by perpetual check occur when one player, usually though not always with a queen, can deliver check regardless of the square chosen by the opponent's king. There's simply no escape from the checks.
(4) Threefold repetition
You or your opponent can claim a draw if the same position occurs three times in the same game, all with the same player to move. Such repetitions can easily occur in endgames when one player is checking another. To claim such a draw, you need to have an accurate scoresheet and demonstrate to the tournament director that the claim is valid. In one game between Fischer and Spassky, Fischer incorrectly claimed such a draw but Spassky, sure that Fischer must have been correct, agreed to the draw!
(5) Insufficient mating material
If neither side has sufficient material to win, either player can claim a draw. For example, suppose that each side has a king and a bishop. Try as you will, there's no way to FORCE a win. You can win force checkmate with a king plus a bishop and a knight (though some masters have trouble doing that!), but there is no way to FORCE checkmate with a king and two knights.
(6) The 50 move rule
This last rule is merciful. If after 50 consecutive moves by both players, no pawn move or capture has occurred, either player can claim a draw. Obviously, you will need to have kept an accurate score sheet to make this claim!