As it has been said, the top competition is Euroleague. This year it is converted to a 16 team league format. After 30 matchdays (teams play with each other in home and away), around April, best 8 teams enter the playoffs phase. Playoff winners attend the NCAA style Final Four, win or go home semifinals and final. Last year CSKA Moscow won, Fenerbahçe İstanbul (which I support) was the runner up, Baskonia and Lokomotiv Kuban were the other two. 4 teams eliminated in playoffs were Real Madrid, Panathinaikos Athens, Barcelona, and Red Star Belgrade. This year 4th matchday has been played with 16 teams. For more info:
http://www.euroleague.net/There are also more pan European leagues like Eurocup, Eurochallenge, FIBA Europe Cup but the level of play drops drastically as best teams in the continent play in Euroleague.
So which teams attend above pan European leagues? Teams from different countries, thus the national leagues. All teams in above competitions play at the same time, all season long, in their national leagues. It can be said Spanish and Turkish leagues are the best in terms of overall quality. Greek, Italian, Russian, German teams follow them in no particular order.
There are also Regional Leagues. The most prominent one is Adriatic/ABA league, where Zizic also plays some games. This league features good teams from all around Balkan region. Why teams attend this regional league? Let me illustrate.
For example, Red Star Belgrade plays in Euroleague albeit with a small budget and homegrown talents. But the quality of Serbian league is really low. So they attend the ABA league to play games not in level of Euroleague, but better than Serbian league. They play in 3 leagues: Euroleague, ABA league, Serbian league at the same time. Of course there is also traditional rivalry between some Balkan/ex-Yugoslavian teams and that is another incentive to play there.
All in all, Euroleague is a good starting point to get a taste of European basketball with its best players, prospects, coachs and fans. I say fans because even though I am used to it US fans are shocked (mostly positively, and sometimes rightfully negatively) when they see European fans.