A board position, or state, is a specific layout of the pieces on the chess board. The state space of chess is the number of board positions we can get to from the initial board state. This number is larger than the number of atoms in the universe!
Since a particular board position can occur in multiple games, the number of possible chess games is even bigger than the size of the state space.
While good chess playing machines can be built, completely solving the game looks to be very difficult. How could a computer possibly represent more distinct things than there are atoms in the universe?
Current research is attempting to solve "end-games", where the game is mostly played out and only a few pieces remain. The end-games where 5 or 6 pieces remain have been solved, and researchers are currently attempting to solve the 7 piece end-game. Storing the data for the 5 piece end-games requires about 7 GB of memory, and the 6 piece end-games require over a 1 TB.