Mario Party 4 is split up into six main game boards. Each board has its own peculiarities, which will be explained. Note that each board can initiate any of the Mini-Games (with the exception of Story and Etc. Mini-Games), which is why they are listed separately under the appendices.
In Story Mode, the first five boards can be done in any order. Once they're completed, the final board is unlocked, in both Story Mode and Party Mode. Upon the completion of each board, you must defeat the board host in a Story Mini-Game. Completing all of Story Mode under the Hard difficulty unlocks the Expert difficulty.
One of the keys to winning at the boards is to win the Mini-Games. If you find that you are having a lot of trouble with the Mini-Games, you should practice them in Mini-Game Mode, or restart Story Mode on a lower difficulty.