Here is a list of the biggest and best chess websites where you can play online with other chess players, solve puzzles, and more. This list is made up by my preference, so you might disagree with it. But I can assure you that most people probably think the same as me because as someone who has logged in and played on each site, I can say that all of them have their own pros and cons.
Lichess.org is no doubt one of the best chess websites out there, or maybe even the best of all of them. It is well-known as a totally free chess website. It is an open source server powered by volunteers and donations. There are no ads, premium accounts, or a puzzle limitation. All the players on Lichess have the same 'status' because there are no membership offered on the website. If you want to play a quick game, I highly recommend Lichess because of its direct design and simple interface. Almost all top chess players play on Lichess!
Pros: Totally free, friendly interface.
Cons: Rating inflatation in rated games.
Lichess.com main page
Lichess.com playing page
You probably have already heard of Chess.com. Since it has the best domain for chess, it will come first in your search result when you type "play chess online" on Google. Some advantages Chess.com has over Lichess are the access to resources that are good for someone who wants to learn chess. The analysis section has more features than most chess sites. It also contains chess news, blogs, and forums that are more friendly to their visitors. But some of the features are not free, which you need to buy the premium membership to get access to.
Pros: A lot of lessons and tutorials.
Cons: Not totally free.
Chess.com main page
Chess.com playing page
Chess24 is also considered one of the biggest chess websites on the internet. It also mostly contains features that are quite similar to Lichess and Chess.com. The advantages of Chess24 over those two are the free access to chess video series and shows. There are a lot of chess DVD series that are presented by popular chess grandmasters. There is also a unique event called 'Banter Blitz' where players get a chance to play chess with top chess players. Just like Chess.com, Chess24 also offered premium membership to unlock certain features.
Pros: A lot of video lessons from top players.
Cons: Not totally free.
Chess24.com main page
Chess24.com playing page