Tickets go on sale to members first and are usually snapped up quickly, but if any are left they’re made available as a general sale. As a rule this only happens for the league cup, champions league qualifiers/group stage or early rounds of the FA cup. To see a league game you really need to be a red member, which costs about £30 for the season, but if it’s a one-off then you’ll need to get in there quickly for matches that go to general sale. The league cup games are the best bet, as they reduce prices to £10 for all lower tier seats and £20 for all upper tier.
If you go to http://www.arsenal.com/membership/buy-tickets it lists upcoming fixtures with dates for when tickets go on sale to the various membership levels.
Edit: Bah, Gooner beat me to it. Sounds like a chap with connections…