There are different types of chess games at Lichess. The type is based on the amount of time, that the players has available on the clock to move their pieces, and, whether the games is rated or not. The classical games play for 25 minutes or more and is the casual game of chess, that beginners will be familiar with. The corresponce games are much longer with up to 14 days per move. The rapid game plays for 8 to 25 minutes. The blitz plays for 3 to 8 minutes. The bullet plays for less than 3 minutes. The ultrabullet plays for less than 30 seconds. You can also learn more about chess by entering the training program, practicing tactics and solving puzzles.
If the game is rated, then the rating will be recorded. If not, then it is just a nice casual game of chess as you know it from the local café. Lichess uses the Glicko 2 ranking system to mathematically determine the rank of each player in the different types of chess games. This is used by the quick pairing feature to match players against each other. The rank will become more and more accurate as you play against other players. The rank is simply a number, where higher means better. You can check the current rank in the different types of chess games in the profile page of yourself and other players. You can also check the general rank distribution for all players and see, how you relate to other players.
If you want to challenge a player, that you see on Lichess, you can mouse-over the player name and a pop-up window will present options for you, such as chatting and challenging.