I'm not always certain as to when I should offer a draw. I don't think any of my opponents have ever offered me a draw. But, I'm in favor of offering and accepting a draw if I believe the game is truly a draw.
There are three basic ways that a game can end in a draw.
1. Stalemate. If one side has no possible moves, but the king is not in check, then the position is a stalemate. Here is an example of a Stalemate:
2. Perpetual Check. Sometimes, one side can launch an attack, but it proves to be insufficient to give checkmate. However, there is an opportunity to create endless checks from which the opposing king can never escape. I believe that it is usually agreed upon that three repetitive moves constitute a draw.
3. Agreement. At any stage during the game, one player can "offer" a draw.
Their opponent then has the option of accepting, or continuing to try and win the game. Proper etiquette is to "offer" a draw as you make your move. Do NOT interrupt your opponent while he/she is thinking about their own move.
Here is an example of a game that I am currently playing, I am black. I am trying to decide if I should "resign" or continue playing in hope of a stalemate. (I intend to resign after posting this. I don't want my opponent to think that I am seeking help from all of you.) So, tell me how you would have played it. Would you have resigned or continued playing?