In The Oxford History of Board Games published in 1999, scholar David Parlett wrote that there are four types of classical board game: race, chase, space, and displace. The book is out of print (but is available direct from the author as a PDF), so I found this description of Parlett’s categorization in a book by Stewart Woods called Eurogames.
In categorizing these public domain or “folk” games, Parlett (1999) draws on the work of H.J.R. Murray (1952) and R.C. Bell (1979) in describing four types of game, as identified by the game goals: race games, in which players traverse a track in an attempt to be the first to finish (e.g. Nyout, Pachisi); space games, in which players manipulate the position of pieces to achieve prescribed alignments, make connections, or traverse the board (e.g. Noughts and Crosses, Twixt, and Halma, respectively); chase games, in which asymmetrical starting positions and goals cast players in the role of pursuer and pursued (e.g. Hnefatafl, Fox & Geese); and games of displacement, where symmetrically equipped players attempt to capture and eliminate each other’s pieces (e.g. Chess, Draughts).
You’re probably unfamiliar with some of these games (as I was). For race games, Parcheesi is a modern version of pachisi…other examples would be Sorry, Candyland, or Snakes and Ladders. Noughts and crosses is tic-tac-toe; other space games include Go and Connect 4. A modern example of a chase game might be Clue. And as written above, chess and draughts (checkers) are classic displace games. (via @genmon)