Escaping is the practice of abandoning a game on an Internet Go server by disconnecting when one is losing in order to avoid recording a loss in the game. Aside from being exceedingly rude and inconsiderate to your opponent, it usually doesn't work; after a set amount of time, the server will usually dock the disconnector with a loss anyway. It is worth remembering that many disconnections are simple technical malfunctions.
Someone who quits a lost game on a Go server instead of resigning it. This is considered rude because it leaves the game unfinished, thus depriving the opponent of her/his victory.
Servers vary on how to handle escapers, some will forfeit the escaper after 5 minutes, others are more lenient.
My opponent escaped, what can I do?
Wait five minutes, if they don't come back and resume they forfeit. But you must be on the server to get the forfeit.
If your opponent escapes more games (usually 10), you will get a win by forfeit.
You can wait for your opponent to return, or just close the window. If your opponent returns you can click the resume button in the top right.
My opponent is rude and I no longer want to continue, what can I do?
You're pretty much stuck. You must finish the game, resign, or leave (and forfeit).
You should leave the game. Do not resign the game if it's rated, this corrupts the rating system, and is against the KGS TOS.