Lichess
Open source chess server
Biggest open source chess server, also supporting a few chess variants. Uses Fairy-Stockfish for chess variant analysis and play against the computer.
Chess.com
Biggest commercial chess server, also supporting many chess variants. The variants server is a bit slow and unstable, but has a big user base. Within some constraints it also supports playing user defined variants.
Pychess
Open source chess variant server
PyChess is a chess variants website trying to provide the look and feel of lichess. It supports a large variety of variants, including analysis with and play against Fairy-Stockfish. Currently is the only website where all major regional chess variants can be played and analyzed. Not to be confused with the PyChess GUI.
Lishogi
Lishogi is an open source shogi website based on lichess with a similar set of features, supporting normal playing, tournaments, studies, etc.
Playstrategy
Playstrategy is an open source abstract strategy game website based on lichess, supporting some chess variants and additional strategy games.
PlayOk
PlayOk is a free abstract strategy strategy game website supporting a few regional chess variants. It has quite many players, but a rather dated interface.
81dojo
81dojo is a free shogi website, perhaps the biggest not only available in Japanese language.
clubxiangqi.com
Clubxiangqi is a Vietnamese Xiangqi server with a fairly big and strong user base.
greenchess.net
Chess variant website for playing correspondence games in a large variety of variants.
SchemingMind
Chess variant website for playing correspondence games in a large variety of variants.