"The knife is ours."
Translation:Il coltello è nostro.
First note this subtle difference: Here nostro is being used as "ours" - "the knife is ours". Nostro can also be used as "our" - as in "it is our knife"
The knife is ours - il coltello e' nostro
It is our knife - e' il nostro coltello
The article is required in the second case because the possessive (nostro) is directly in front of the noun.
Or perhaps another way to think of it is: The noun is "il coltello" and when you want to say "our knife" you insert "nostro" between the article and the noun. Notice that the first sentence (The knife is ours) still has "il coltello" - it's just that the word order of the sentence has moved around to change the meaning of nostro to "ours"
It's a bit confusing because Italian doesn't use possessives in exactly the same way English does, but I hope that makes sense.
In English, "a/an" are the indefinite articles and "the" is the definite article.
In Italian, "un/una" are the indefinite articles and "il/i/lo/l'/gli/la/le" are the definite articles.
All adjectives, including articles and possessives, agree with the noun they go with. So when it comes to possessives, they must agree with the thing possessed, not who possesses it.
il mio - my singular masculine thing
i miei - my plural masculine (or mixed/unknown) things
la mia - my singular feminine thing
le mie - my plural feminine things
It's a good rule of thumb, but there are some exceptions (no language is without irregularities): http://italian.about.com/od/grammar/a/gender-of-italian-nouns.htm
It's always "il nostro gatto". The possessive adjective requires the definite article (except for singular unmodified family members).
When it's the possessive pronoun, however, you can use the definite article or not, but the meaning changes subtly.
"La gatta è la mia" means "The CAT (and not something else) is mine."
"La gatta è mia" means "The cat is MINE (and not someone else's)."
"Le" and "nostro" never go together because that violates grammatical gender agreement.