"We visit our mother at home."
Translation:Matrem nostram domi visitamus.
Please correct me if I am wrong, but I think I am starting to figure it out: The "our" is classified as an adjective (not as a possesive pronoun like in my native language), it belongs to the object (not to the subject of the sentence) in gender and number, but in form (declension) it is conjugated in its own way ( it does not follow the declension of the object, but is conjugated in its own manner (its own declension). Right?
Yes, adjectives do not change to match the declension of the noun they modify. They only change based on gender, number, and noun case. So, if a noun is feminine, singular, and accusative (like matrem here), then the adjective must agree by being in its feminine, singular, accusative form (like nostram). Matrem is a 3rd declension form while nostram is a 1st declension form.
I should note that some adjectives has forms in both the 1st and 2nd declension (feminine forms are first declension, masculine and neuter forms are second declension). But adjectives that are 3rd declension will always be 3rd declension.