What is Lichess?
Lichess is a free and ad-free chess website, as well as an open-source chess-server that can be run on a server machine. It's very similar to chess.com. It features online play for blitz, bullet, rapid, classical, and correspondence, as well as chess variants including crazyhouse, chess960, king of the hill, three-check, antichess, horde, and racing kings. You can create custom games, and it allows three different options for the timing: real time, correspondence, and unlimited. The website has a great deal of polish, the GUI feels intuitive, it runs very well, and I've yet to experience a single bug or error with it. They also have a mobile app for both Android and iOS. Lichess is completely free; unlike chess.com, you do not pay for more features.
What features does Lichess have?
All the points below are about the Lichess website (and app). I assume many of these options are offered in the open-source server software. I haven't run the server software on my own machine and probably never will, so I can't speak for it.
1) A rating system
While there exists chess software that analyzes your chess performance and gives you an estimated ELO ranking, Lichess really shines because your rating is determined solely by your playing rated games with other players (as opposed to against the software). At first, you're assigned a provisional rating of 1500, and you have to play many rated games in order for your rating to stabilize and lose its provisional status. The provisional status is removed when your "rating deviation" drops below 120. I believe the rating system is very similar to ELO rankings. I have seen on the forums that Lichess ratings are 100-200 higher than FIDE ELO ratings, though.
2) Puzzles
There are a ton of chess puzzles that you can play. There are what I believe to player-made puzzles, in which you have to "find the best move". Sometimes, you will have to find multiple consecutive "best moves". I don't always agree with the best move, but in general they help you think and train. They have ratings assigned to them. There is also an official "practice" mode in which you can practice various checkmates, pins, skewers, smothered attacks, as well as intermediate tactics and pawn endgames. I haven't used them as of yet (mainly because I haven't taken the time, I'm sure they're pretty good.)
3) Player-made studies (and the ability to create your own studies)
Analysis of chess games beyond simple computer analysis; you get to see the human-written commentary move-by-move (I believe you can also play the study boards.) Some of them are really good, some of them are garbage, some are between the two. I personally don't like them but I see nothing wrong with the fact that they're an option. I wish you could choose to browse chess commentaries by the author's rating. I really don't care to read or watch a commentary by someone who doesn't have a rating over 2000. You can sort them based on age, popularity, and "hot" (I don't know how "hot" differs from "popularity" and they don't really explain it).
4) Coordinate training
The software displays an empty chessboard, and coordinates pop up on the screen. Your job is to click on the appropriate tile as quickly as possible within the given time limit. Once the time has elapsed, you can view how many "points" you accumulated (that is, how many coordinates you got right). It also shows your average score. You can play coordinate training from black's perspective or from white's perspective. I play it a couple of times a week, and it's helped a little bit. I'd like to commit to it every day though.
5) Tutoring coaches
The Lichess website has a "tutoring" tab, in which it displays various tutors, their tagline, their languages, their availability their chess rating, and sometimes their fee/rate. You can click on their profile to see their "about me" information, playing experience, teaching experience, "best skills" whatever that means, and teaching methodology. You can also see reviews on their profile, though I can't speak for the accuracy of the reviews. From the "coaches" page you can choose to sort the various tutors by when they were last online, their rating, and how many positive reviews they have. I'm not entirely sure how the meetings happen, I assume you and the coach would have voice or video chat and play an online game simultaneously, at which point he can walk you through various tactics and weaknesses.
6) Tournaments
Tournaments take place on Lichess. Some of them are just-for-fun, some of them have a cash prize pool, some of them are official tournaments for chess organizations (I believe this year's USCF championship was hosted on Lichess due to the pandemic), and some of them are hosted regularly and the winner has to defend their title in the next tournament. Many tournaments are restricted to people who have a 2000+ rating.
7) Game streaming
You can watch ongoing games between GMs (grandmasters), IMs (international masters), FNs (FIDE masters), and LMs (Lichess masters) natively within Lichess. You can also watch games taking place in tournaments, you can browse online streamers (which generally links to their Twitch. I didn't know there were Twitch chess streamers. The more you know.) You can also select and watch formerly-streamed games in the form of videos. I don't know if every broadcasted game is stored as a video, or just some arbitrary "select games" are chosen to be videos.
8) Various tools
Much like other chess software, you can run an analysis board on all your games, in which it lists all the moves on the side and you can walk through your games move-by-move, which is helpful. All the games you play are also stored on your profile, which means that you can view them and pull up the analysis board at any time. The chess engine Stockfish is available to play against at various arbitrary "difficulty" levels, which I suspect is just its depth configuration. The bot's estimated rating per difficulty level is also listed. I can't speak for its accuracy but I'm guessing the Lichess people knew what they were doing. There is also a "board editor" available, in which you can click and drag pieces anywhere on the board and export the board as a FEN string, as well as play the board from that point onwards. You can also import a game based on a FEN string. There is also an "opening explorer" option but I am not familiar enough with it to know how it works.
9) Profiles (and online play)
I've stated this already, but there is online play. You can create an open game with various options attached to it (such as time, rated or casual, type of game, etc) and then people can browse all open games and choose the one they want to play. People can follow your profile, and you can follow their profile; you can invite your friends to Lichess and play against them. All your rankings, game-stats (wins/losses), etc. are listed on your profile, as well as a short bio that you can put text or a website into.
How does this compare to chess.com?
Chess.com has many of the same features as Lichess, but I believe it has higher-quality puzzles, as well as more chess engines that you can play against (as opposed to just Stockfish.) However, in order to unlock the analysis board, extra chess engines, unlimited chess puzzles (as opposed to limited), chess training, and chess videos, you have to pay for a monthly subscription. Lichess has chess videos, the analysis board, and unlimited puzzles for free. Obviously I'm biased to prefer Lichess, but in my opinion the premium version of chess.com is marginally better than Lichess, and the free version of chess.com is marginally worse than Lichess. Do with that what you will.