Is there a way to play the mafia party game online ?
In addition to sites dedicated to mafia like those already described, a lot of forums that have a "Forum Games" section will be open to running mafia games if they aren't already - I play mafia regularly on http://forums.xkcd.com and have seen it being run on many other forums I frequent.
I'm playing right now at http://mafiacity.org the game is just amazing!
Gameplay is amazing. Probably the best mafia mmorpg game online I ever played!
Brainden.com is another great place to play for those just starting out in the online mafia party game world. The games are in the Forum section. Mafiamaniac.net runs games that are a little more advanced, but the membership is very active and helpful, even providing mentors for those who haven't played before. Here is a link to a mafia introduction/guide to playing that you may find helpful if you are new(ish) to the game.
Anyhow, I wouldn't want to play on twitter, but if you search there are whole on-line sites dedicated to the likes of werewolf/mafia, so yes. Have fun.
This link is more useful than the wikipedia one:
The links at the end are what you're looking for.
There are also Mafia games played in forums like mafiascum.net.
http://www.gamerz.net/pbmserv/ hosts many games, abstract and social and otherwise, including Werewolf, via email.
You can play Mafia live, over webcams, at Daily Mafia.
Since no one else mentioned it: Mafia is, and has been, one of the top 10 Starcraft II Arcade maps for several years.
They added a lot of new features over the face-to-face game, including many that wouldn't be possible in-person. And since SC2 is so popular, it usually only takes a minute to find a full party of 12 people.
The game itself is $30 (sometimes you can find it on sale for $10), but you can play Mafia for free if you play with a friend who owns the game.