A "WIN" bet is just what it sounds like: betting that the horse you pick will win the race. You win if the horse wins.
For a horse to "PLACE" it must finish the race either first or second. This bet generally pays less than the win bet but it gives you two chances to win instead of just one.
For a horse to "SHOW" it must finish the race either first, second or third. This bet generally pays less than either the win or place bet, but it gives you three chances to get a return for your investment.
ACROSS THE BOARD
This is a quick way to say that I want to bet a horse to finish in any of the first three positions. ($2 Across The Board = $2 to win, $2 to place, and $2 to show). If the horse wins you collect all three bets. If the horse finishes 2nd then you collect the place and show bets. If the horse finishes 3rd then you will only collect the show bet.