The options menu allows you to change the settings on your account, and optionally activate various privacy settings. You should also always use the options menu to logout after playing.
The 1st option allows you to toggle the camera between 'auto' and 'manual' mode. In auto mode the camera will attempt to automatically rotate so that your character is always visible.
The 2nd option allows you to toggle between using two mouse buttons to control the game, or just one. In single button mode the menu will always appear when you click on more than one item. Some people find this easier, and it's also handy if your mouse only has one button!
The 3rd option allows you to change your password. It is a good idea to choose a long password which other people will not be able to guess. You should also NEVER tell other players your password for any reason, the Runescape staff will never ask you for your password, so if someone asks don't give it to them. It is also not a good idea to use the same password in more than one place, in case someone manages to guess one of your other passwords.
The next 3 options are privacy settings. They are applied to all people who are not on your friends list. You can use these to block chat and/or trade messages from all people who you haven't listed as friends.
The privacy options can be used to prevent you from being shown as online on other peoples friends list when you login. Once activated only people you have listed as friends will be notified that you have entered or left RuneScape.
The final option logs you out of the game. You can't log out whilst fighting, or just after combat, but at all other times you are allowed to leave. When quitting runescape you should always use the logout feature here to ensure your player is saved properly ready for next time you play.
RuneScape Wiki note: the text starting from 3rd option could vary as the original document is unavailable. To keep continuity in the article, the text from 2002 was carried over.