It's good that you're aware of the concept of agreement.
Agreement of noun and adjective means that a noun and the adjective modifying it have to match, in both gender and in number. Since "perro" is masculine and singular, the adjective must match and also be masculine and singular: "nuestro."
I think what may have been confusing you was the S at the end of "ours." Perhaps if the English sentence had been "Bobby is our dog," then you would have been able to go directly and correctly to "Bobby es nuestro perro." So I'll explain that "ours" with an S on the end is a possessive pronoun, whereas "our without the S is a possessive adjective. In Spanish, for the first person plural, the same word works for both types of possessive. Here's the table in English (the possessive adjective is listed first, followed by the possessive pronoun):
his, his (there's no change because the first one already has an S)
its, its (")
In Spanish we will need two such tables, one for singular:
[vuestro/vuestra, vuestro/vuestra, if you are in a vosotros country]
and one for plural:
[vuestros/vuestras, vuestros/vuestras, if you are in a vosotros country]
You didn't ask about this, but since I've gone this far I better finish the story about possessives. In "El perro es nuestro," strictly speaking nuestro is still an adjective. However, possessive adjectives used in that way in a sentence will follow the table I gave you anyway. Thus, in "El perro es mío," mío is an adjective, but we do need to pull the second entry from the table, because the possessive is stressed in that sentence.
Perhaps more information that you needed. Back to the main point. I'm guessing that the S on the end of "ours" threw you off and led you to think you needed to add an S to "nuestro." However, the thought process should be, first check if the noun is singular, and whether it's masculine or feminine; also check whether your possessive is stressed or unstressed; then pull the entry that agrees in number, gender and stress status out of the table.