Here are the biggest most important chess websites. I will give the cost for a one year full membership where applicable. Most have other purchase options. They are listed in no particular order, and I do not claim that this list are the only sites to be recommended.
Pay To Play:
Chess.com : It's the biggest. To enjoy all the features you will have to pay. Still there is a lot. Matches of just about every kind. Tournaments, news, puzzles, problems, computer play, bots, articles, and lessons. If it has to do with chess they probably have it. Currently $99 for a year of full access.
Chess 24 : Another big site with lots to offer. Again to have full access you will have to buy a membership. 1 year membership at the time of this writing $149.99.
Internet Chess Club : AKA ICC Has been around a long time in internet years. Another big site with a lot of content. USCF holds rated tournaments through this site. 1 year $69.95.
Free Chess Sites:
Play Chess : By Chessbase. This used to be a paid site but now appears to be free to non professionals. Chessbase is the professional chess database software company. Also home to Fritz, and other chess engines. Certainly worth checking out.
FICS Free Internet Chess Server : Before there were chess websites you could download an interface and play over a chess server. Both the ICC above and FICS date back to those days. You can still download a GUI Graphical User Interface and play that way. Also you can play through your browser. It's free. "We do it for the game not for the money". Appears to be their moto.
Lichess : Last but not least is Lichess.org. "lichess.org is a free/libre, open-source chess server powered by volunteers and donations." The name is derived from live/light/libre chess. It is another full featured website with game to play, watch, or analyze, videos, etc. I love the look and feel. The analysis graphics are excellent. It compares well against the paid sites and it's free.
Others: This category I could use some help filling if you know of a site to recommend please leave a note in the comments.
Decode Chess : Using Artificial Intelligence AI to analyze your games and explain why a move is good or bad. If you haven't tried it you really should. I find it interesting.
Chessable : A chess teaching platform. Using advanced teaching methods. You can use many free courses and samples and purchase full blown chess lessons.
Bill Wall's : A huge research and links site. Looking for pgn game collections, and books, this is the first site to look at.
Well that's all for now. Let me know what you think. Leave a comment / suggestion.