Russian has a special kind of adjectives called "possessive adjectives", they mean belonging to somebody and answer the question whose? The possessive adjectives are formed from a noun which names the possessor person and are used only in their short adjective form, they don't have the full form, which means they are not declined like nouns, also they agree in gender and number with the possessed noun.
If the possessor noun ends in -a/-я (feminine or masculine gender noun, 2nd declension nouns), the possessive adjective is formed with the suffix -ин which substitutes -a/-я:
бабушка – бабушкин 'belonging to grandma': бабушкин дом, бабушкина комната, бабушкино платье, бабушкины сказки
If the possessor noun ends in a consonant (masculine gender noun), the possessive adjective is formed with the suffix -ов:
брат – братов 'belonging to brother': братов велосипед, братова комната, братово слово, братовы книги
Nowadays, the -ов possessive adjectives are rarely used, although the -ин ones are pretty common, especially those made from personal names, like сашин (belonging to Саша), колин (belonging to Коля), наташин (belonging to Наташа), андрюшин (belonging to Андрюша), танин (belonging to Таня). Russian adjectives, even if formed from personal names, are not capitalized.
The possessive adjectives compete with the Genitive case of the same nouns, the meaning being pretty close, belonging to somebody vs. somebody's:
бабушкин дом — дом бабушки
сашины друзья — друзья Саши
It can be easily noticed that the Russian surnames ending in -ин and -ов descend from possessive adjectives. Петров 'belonging to Пётр (Pyotr), Pyotr's', Ильин 'belonging to Илья (Ilya), Ilya's'. These possessive adjectives got substantivized (became to be perceived as nouns, they answer the question who?), but still kept their adjectival case and gender forms.
That said, it is clear that Пушкин means 'belonging to Пушка (Pushka)'. Пушка (Pushka, 'Cannon') was the nick-name of one of Pushkin's ancestors.
Пушкин, like all the rest of the possessive adjectives, is declined as a typical short-form adjective, in the plural (the Pushkins) it has the following forms:
Nom. Pl. Пушкины
Gen. Pl. Пушкиных (like красных)
Dat. Pl. Пушкиным (like красным)
Acc. Pl. Пушкиных (like красных)
Instr. Pl. Пушкиными (like красными)
Prep. Pl. (o) Пушкиных (like (o) красных)
That is why, 'the house of the Pushkins' is дом Пушкиных.
From a surname, a real full-form adjective can also be formed, but in this case the meaning is different, it means 'of or related to ', like in English: Shakespeare — Shakespearean 'of or related to Shakespeare'. With Пушкин, such an adjective is пушкинский meaning 'of or related to Pushkin, Pushkinian'. Examples:
пушкинский музей — 'museum dedicated to Pushkin'
пушкинские места — 'places connected with Pushkin's life'
Пушкинский парк — 'park named after Pushkin'
