Super Mario Brothers 3, alternately known as Mario 3 or SMB3, is arguably the best Game in the series. Produced by game design mastermind Shigeru Miyamoto, SMB3 melded the lessons of the two true previous games* in the series, and added upon them.
With the best level design in the series, with a then-innovative map-screen interface that allowed for inventory, to the range of colorful and iconic power-ups that so trapped the minds of many at the end of the 80's, SMB3 had it all. The game control was superb, and as is Miyamoto's hallmark, replay value was high with both incredible amounts of secrets per level, and an inclusion of the original Mario Bros. Game (from before they were 'super' ;) for the two-player option.
Truly one of the best. The Super Nintendo versions, although good, seemed to expand laterally upon the concepts introduced in SMB3, instead of the massive jump from SMB2 to 3.
And besides, Yoshi had nothing on the Koopa Clan that never saw light of day in its unabridged format again.
*you should probably hear The Truth Behind the Super Mario Series in America