I recently watched a game that ended something like this.  When the white king was put in check, would a move to the b4 square be a legal move since the black bishop on e7 is pinned?  Please explain or direct me to a rule on this please.  Clearly, the rule states the king can't put himself in check, but would the king BE in check if that attacking piece is pinned?  Thanks for the help.

It's still illegal.

Think of it this way. If it loses your king in bughouse, it's illegal.

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You're correct, of course, but what does that phrase mean? "If it loses your king in bughouse"?

http://en.wikipedia.org/wiki/Bughouse_chess

That a piece is pinned has absolutely no effect on it's ability to give check. Moving 1. Kb4 is illegal in the above position. There have been instances in which a pinned piece delivered mate through discovery, when the player simply moved his intervening piece.

FIDE Laws of Chess are quite specific:
3.9    The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained from moving to that square because they would then leave or place their own king in check. No piece can be moved that will either expose the king of the same colour to check or leave that king in check.