I want to make a chess-playing site for profit is it legal for me to use open source software to make profit?

The term ‘Open source’ generally means that the source code is available to developers and designers to use and expand. Really, to answer your question you have to read the fine print of the open source agreement. The “strings” attached could be anywhere from ‘this is for personal use only, and if you make money on it we get a cut’ to ‘just give us credit and include our readme file in your docs and you can do whatever you want.’

Really, the only legal definition of an open source agreement is what you agree to when you download or begin using the software. I would go back to that and read it carefully. If you are serious about making money off anything designed be someone else, you may also want to consult an attorney that deals with such matters.