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.
Hey @Lynob A few days ago I released RiseTournaments app(Reversi real time game) , I organise weekly tournaments with great prizes for top 3 players . It's FREE and available on App store. If you are interested feel free to try it and give us feedback :)! here's the link: https://itunes.apple.com/us/app/risetournaments/id942714819
Kacper's answer "Kurnik" is only for polish. The international version of the same site is playok.com. If you are reading this in 2018/2019, the app called othello quest on android and iphone is the place to play with real players. No I'm not a creator of these, I'm just an expert player who knows where most of the good players play.