I was very good at English, but I had to look up part of this in my high school English Handbook--it's been several years.

"I" = subject.
"saw" = verb.
"him" = direct object of "I."
"do ten push-ups" = prepositional phrase that functions as an adverb of "saw."
(Note: "ten" = adjective of "push-ups").

Also, the mood of "do" is indicative. Here's a quick reference:
Indicative states a fact or asks a question.
Imperative gives a command or request.
Subjunctive expresses an action or condition as a conception of the mind rather than as a reality (i.e., "If I were you. . .")

Further note: the link gremlinn gives is correct as far as it goes, but it focuses on what I feel is not the main point. In any sentence that uses the pattern "I saw," a large part, if not all, of whatever comes after the object will function as an adverb.

Further further note: the word "do" is in the present tense. The explanation is a little convoluted, but here goes. It's because even though the speaker is talking about something that happened in the past, when he actually saw it, it was happening in the present for him.

Anyway, I hope all this helps.