This time Bowser has got himself a time machine and has traveled to different times in the history of mankind and stolen something important to have in his own personal museum back home.
Yoshi wanders of into Bowser's museum and is captured by Bowser. Mario runs after Yoshi into the museum but before he can save Yoshi he must return all the things that Bowser has stolen.     Inside the museum there are multiple doors. Behind each door is a version of the original Mario Bros game. Only this time Mario can't be hurt by the enemies that come out of the pipes. When you have defeated three Koopa Troopas you are rewarded with one item that belongs somewhere in time. After that you can enter the time machine standing in the middle of the room and choose from many different dates to go to. Here follows some examples:
80 million BC - the Dinosaur age
776 BC - Greece, the Olympic games
31 BC - Egypt, the Pyramids
1879 - Thomas Edison's home
1602 - Shakespeare
1947 - India, liberated by Gandhi
1969 - The Moon and the Apollo project
    The different times are presented as very short platform courses with weird hourglass or clock shaped blocks all over the place. If you jump under special info blocks you receive small pieces of information about the current time. From that information you must think of what item to bring to this time.
Each of the "Mario Bros rooms" contains a unique item and you can only carry one item at a time, so you'd better memorize them. If you e.g. find a egg in one of the rooms you go to the Dinosaur age and return it there. In 776 BC you'll need to return a torch and so on.
    The enemies you encounter are mostly Koopa Troopas, some enemies that could be Hammer Brothers (they don't throw their hammers) and a weird onion like walking enemy.
You can't be hurt by these enemies either, you just get "stunned" for a short while.
There are also human friendly characters walking around on the courses. On the Apollo course there are Astronauts and on board Magellan's ship there are sailors. When you return a item you are thanked in the newspaper (!), I don't know if they had a newspaper in the dinosaur age, though. ;-)
    In the SNES version Mario travels through time by surfing around on the big "Ocean of time" on a surfing board in glorious Mode-7. The different points of time that he can visit is represented by small whirlpools in the ocean! The NES version, I guess couldn't handle the Mode-7 effect so that was removed when converted to the 8-bit console...
Note: the facts about this game is based mostly on the NES version of the game.

   

Bowser has caught Yoshi.
 

A Bowser statue (of course it is _Bowser's_ museum)
 

The Mario Bros like sub-game
 

All the different times that you have to pass in order to rescue Yoshi.
 

The Dinosaur age
 

Mario encounter of the third grade.
 

You are thanked in the newspaper for returning the torch in 776 BC, Greece...