Ever since the 1980s there has always been a fair amount of choice in relational database management systems. But since the turn of the millennium, we now have ample choice in the open source world too. In fact, some of the strongest RDBMS tools are open source.
Below is a listing of my recommendations of good RDBMS systems and tools. It is biased toward free and open source software. Since it is an opinionated list by a single person, there are bound to be good resources that I have missed. Nonetheless, I hope this serves as a good starting point.