Most Sudoku puzzles published have only one solution. If there is more than one solution, it is probably a mistake. That said, puzzles with incomplete clues can have multiple solutions. In the extreme case, a puzzle with no clues has 6,670,903,752,021,072,936,960 solutions according to Wikipedia.
I don't know if it's possible to have exactly 3 solutions, but boards with 2 and 4 (and more) solutions are easy to find. In general, I think boards with an even number of solutions are easier to create.
Finding the number of solutions is a generalization of Sudoku solving algorithms, and there are Sudoku algorithms that do significantly better than brute forcing. Once the board has been filled out as far as possible, it can be brute forced the rest of the way.