Welsh rugby is controlled by the governing body the Welsh Rugby Union or WRU (boo hiss!) which oversees 320 member clubs, the Welsh national team and National Leagues and Cups.
The current president is Gerald Davies.
Currently (And I say currently because fuck knows what the WRU will do in the coming months.) Welsh rugby has four professional sides that play in the PRO14 these are:
Cardiff Blues/Gleision Caerdydd:
Formed in 2003 they play at Cardiff Arms Park in Cardiff. They have had some success and have twice won the European Challenge Cup.
Current squad is:

Dragons/Dreigiau:
A.k.a. The Cows they formed in 2003 and play at Rodney Parade in Newport. Their success is less measured in trophies and more measured in how many careers have been rebuilt and the amount of lives saved.
Current squad is:

The Ospreys/Y Gweilch:
Formed in 2003 they play at the Liberty stadium in Swansea. They have had some moderate success winning the Celtic League four times in 2004–05, 2006–07, 2009–10 and 2011–12, bag of shite in Europe though.
Current squad is:

Scarlets/Y Sgarlets:
The Scarlets or The Turks as they are also known formed in 2003 and play at Parc y Scarlets in Llanelli. They have also had some success winning the Celtic League in the 2016-17 season beating Munster 46-22 in the final.
Current squad is:

Below this are several league where semi-professional and amateur clubs play.