What is a "Power Throw," and who all gets one?

There is nothing that really defines "Power Throw" in SSB. The reason they call Ness's back throw a power throw is that it is very powerful, high KO potential. Here is the entry from the wiki:

Throws opponent behind him with PSI powers. Has the highest KO potential of all throws.

Power Throws is a community driven name given to throws where combos and knockups are not possible. Power Throws are used to either KO opponents or do throw them off the stage so that they have to go into recovery mode.

Here is a good example of different throws: Mario's Up Throw is a great combo potential throw. Mario's Up Throw will never KO an opponent, so it will never be called a Power Throw. His Forward and Back throw are Power Throws, because it launches an opponent in said direction, either wanting a KO or forcing his opponent off the stage and forced to recover.

If you want to know more about throws, here are the links to them.

Forward

Back

Up

Down