The first two responses above were correct. The third - mine - was simply an attempt to put some flesh on the bones, as it were, by illustrating how the two words, past/passed, worked. It seems that was not enough, so here's some more.
An adverb qualifies - ie tells a little more about - a verb. Thus, in 'He walked slowly', 'slowly' tells us how he walked. In 'He walked under the ladder', 'under' tells us where he walked.
In exactly the same way, in 'He walked past the post office' , 'past' tells us where he walked, so 'past' here is an adverb. The same thought could be expressed purely verbally without an adverb just by saying, 'He passed the post office'.
So, 'He walked past the post office' is correct and 'He walked passed the post office' is a non-starter!