Final Smashes were introduced in Super Smash Bros Brawl as it's main new mechanic added to the gameplay. One can perform a Final Smash by breaking apart a Smash Ball when it spawns. Once you've broken the Smash Ball, you can perform a Final Smash at any time by doing the same button input as you would a neutral Special Move, but you have to use it quick, as someone could knock the Smash Ball back out of you with an attack and take it for themselves. Each Final Smash is unique to each character, or at least in most cases, save the Landmaster shared by Fox, Falco, and Wolf, as an example.
A transformation is the most typical of final smashes, being one of the most common in Brawl and very unoriginal, generally just making the character have more damage/knockback/invulnerability for a set period of seconds. MYMers prefer to use Final Smashes to bring the mechanic or play style of a moveset together, an excellent example being the Final Smash of Snorlax.
Final Smash Example:
Bowser's Final Smash. Flee in terror, mortal fools!
Bowser's Final Smash: Giga Bowser
A flash of light occurs over Bowser and he turns into Giga Bowser from Super Smash Bros. Melee. Giga Bowser takes no knockback from hits, having constant superarmor, but still takes damage as normal. Foes will still be hard pressed to damage Giga Bowser however, as due to his new massive size (Around three times that of his regular size), Giga Bowser can hit foes from just about anywhere, and you certainly don't want to be hit due to Giga Bowser's massively enhanced damage and knockback. Lasts about 15 seconds. Moves to use with this move to actually hit your enemies with include your up special, dsmash, and usmash.