What is the appositive phrase in this sentence, “Philip is walking with Rufus, an old English sheepdog.”?

Hint: An appositive noun or pronoun phrase comes after another noun or noun phrase in apposition to it, providing knowledge that helps to identify or describe it further. Unless the appositive is restrictive, certain "bonus facts" are framed by commas (i.e., provides essential information about the noun).

Complete answer:
Appositives are nouns or pronoun phrases that rename the noun immediately next to them. They may usually be removed and the sentence will still make grammatical sense, but they provide meaning and additional detail.

The knowledge provided by an essential appositive phrase is necessary for defining the noun or pronoun that comes before it. The sentence would be meaningless without the crucial appositive expression. A nonessential appositive expression, on the other hand, adds to the sense of a noun or pronoun in a sentence whose meaning is already clear. It provides the reader with additional—but non-essential—information.

Example: Reenu stood on the street with her brother, Will.
In the above sentence, appositive is Will which renames the noun brother.
Now, in the above sentence given in the question, the appositive word renames the noun 'Rufus' after an Old English sheepdog.

Note: Apposition is a grammatical construction in which two objects, usually noun phrases, are positioned next to each other, with one identifying the other in a different way than the other. The two elements are said to be in apposition, and one of them is known as the appositive, but determining which one is the appositive involves looking at how the elements are used in a sentence. Apposition is a scheme figure of speech that occurs when supporting clauses' verbs are removed to produce shorter descriptive phrases. Since they can interrupt the flow of a sentence, they are often used as hyperbatons, or figures of disorder.