You can start a private match with another player by doing /1v1 <playername>. You'll be able to choose the 1v1 arena (or a random one!) and send them a request. It's up to them to accept or deny though, so send them thoughtfully!
What do you mean I gained/lost a point after a 1v1?
You earn points for each element as you win (or lose!) 1v1s. If you are a water bender and win a match, you earn a point for water. If you're earth, you win a point for earth. Same for losing points.
Your points will only change if you're close in /b level to the other player, so fight people near your level.
People keep interrupting my 1v1 in the general arena!
That can be annoying but always remember the normal arenas are free for all arenas! You can ask people to leave your 1v1 alone, but things happen. If you don't want to be interrupted (or accidentally hit anyone), make sure you use /1v1