user8291: My understanding is that there are basically two ways in which a game of chess is won.
(1) One side gives (or forces) checkmate, regardless of material situation on the board. He may be in material inferiority, but mate is unavoidable.
(2) One side achieves enough material (or positional) superiority that with correct play will inevitably lead to a material advantage (if superiority was positional) won endgame and then checkmate, although at this stage mate is hardly forced. But the other side, for reasons above explained, resigns. 
Am I missing something? Are there a 3rd, and maybe even a 4th, case (not considering forfeits, illegal moves, and so on...), in which a game is won?

user12629: If you don't count forfeits (including time forfeits) then yes, the two possibilities for losing a game are resignation or checkmate.  There may be other reasons for resignation than material or positional inferiority, however.  
In 2017, GM Hou Yifan resigned after 5 moves in protest over perceived manipulated pairings.  (The pairings were later analyzed and proven to be correct.)
One time I was down an exchange in a tournament game and the police wanted to talk to my opponent. He didn't resign (the TD paused the clock) but had the circumstances necessitated, he might have resigned to go take care of that matter, despite his material superiority.
And, of course, a player might resign without being inferior because they incorrectly believe they are inferior. Even strong players will occasionally resign a game that should be a draw or even a win for them.

8291: <p>My understanding is that there are basically two ways in which a game of chess is won.</p>

<p>(1) One side gives (or forces) checkmate, regardless of material situation on the board. He may be in material inferiority, but mate is unavoidable.</p>

<p>(2) One side achieves enough material (or positional) superiority that with correct play will inevitably lead to a material advantage (if superiority was positional) won endgame and then checkmate, although at this stage mate is hardly forced. But the other side, for reasons above explained, resigns. </p>

<p>Am I missing something? Are there a 3rd, and maybe even a 4th, case (not considering forfeits, illegal moves, and so on...), in which a game is won?</p>
 12629: <p>If you don't count forfeits (including time forfeits) then yes, the two possibilities for losing a game are resignation or checkmate.  There may be other reasons for resignation than material or positional inferiority, however.  </p>

<p>In 2017, GM Hou Yifan resigned after 5 moves in protest over perceived manipulated pairings.  (The pairings were later analyzed and proven to be correct.)</p>

<p>One time I was down an exchange in a tournament game and the police wanted to talk to my opponent. He didn't resign (the TD paused the clock) but had the circumstances necessitated, he might have resigned to go take care of that matter, despite his material superiority.</p>

<p>And, of course, a player might resign without being inferior because they incorrectly <em>believe</em> they are inferior. Even strong players will occasionally resign a game that should be a draw or even a win for them.</p>
