Othello has been around for 30 years or more, making it a "classic" board game. It is a fairly straightforward strategy game for two players and a great introduction to strategy games in general. The 64 identical pieces are two sided-- one black and one white. Players take turns putting down a single new piece and "capturing" the other color between the piece they put down and one already on the board of their color. "Captured" pieces are turned over to expose the player's color. The game board is ever changing and moves that capture many pieces in one turn can spell disaster in the next.
Buried in the rules is way to help even the game out between more and less experienced players. It turns out the corners of the board are important because they cannot be captured once played, so placing four pieces of the weaker player's color in the corners gives that player some advantage.
My son began enjoying this game around age eight, but his six year old sister can also play. Beginning players first learn how to logic out which spots allow the most captures, later players start to think ahead and sometimes take lesser pieces for greater pieces later.
Although I think part of the fun of the game is the physical act of flipping the pieces back and forth, there are also free versions on the web.