It's called an "objective complement", because it completes the direct object "my dog".
You probably know about subjective complements (predicate nouns, pronouns, and adjectives), which modify the subject and follow a copula verb: His name is Joe. My dog seems sick today. My car is all I have our left.
An objective complement, which can also be a noun or adjective (theoretically also a pronoun, but I'm having trouble coming up with an example), follows the object (usually direct), modifies the direct object, and completes the meaning of the sentence or clause:
I called my dog Rex. I painted my car red.
The objective (and subjective) complement should be distinguished from the so-called apposition, which merely provides extra detail, but doesn't really complete the meaning of the clause:
X We met Bill Gates, the richest man on Earth! (And Steve Jobs, his competitor, was also there.)