Identify board game on 4x4 board with 8 black and 8 white pieces
That looks like a set of Quarto to me. There are eight dark/eight light pieces, eight tall/eight short pieces, eight round/eight square pieces and eight pointed/eight flat pieces. Usually the fourth attribute is hollow/solid, but the game works in exactly the same way with pointed/flat.

To play the game start with all the pieces off the board, and the first player chooses a piece for the second to place. After it's placed, the second player chooses a piece for the first player to place. Play alternates in this way until a player places a piece that creates a set of four in a straight line that all share a characteristic (e.g. four round pieces in a row). The first player to make four in a row wins.