Are there any online big communities similar to FICS or Chess.com for Othello (Reversi) where you can play against many players and so on?
You can play online on Yucata http://www.yucata.de/en/GameInfo/Reversi. Generally the games are played over days rather than "real time" though.
Lots of other great games too
You can try Turn The Tides (I'm its developer) which has 24 different boards based on the idea of reversi. You also get to use powers and there are another 24 boards with hexagonal tiles as well. You can play offline(against the iphone) or online with real players via GameCenter.
My default site for real time gaming is Board Game Arena. It hosts over 200 games. see BGA:Reversi
A few days ago I released RiseTournaments app (real time Reversi), I organize weekly tournaments with great prizes for the top 3 players. It's free and available on the App store.