Learn Spanish (Latin America): Nosotros necesitamos unas sillas pequeñas aquí.

Because the article has to match the word it goes with! Sillas (chairs) is a feminine plural noun, so the word for "some" must also be feminine and plural, which is unas. If we were talking about a masculine word like libros (books), we would use unos.

Does unas always translate to "some"?

Yes, unas (and its masculine partner unos) usually translates to "some" or "a few." It is simply the plural version of una (a/an), which changes the meaning from "a chair" (una silla) to "some chairs" (unas sillas).

Why does the word for small (pequeñas) come after the chairs (sillas)?

In Spanish, describing words (adjectives) usually come right after the thing they describe. So instead of "small chairs," we say "chairs small" (sillas pequeñas). Notice that pequeñas also ends in -as so its gender and number perfectly match the feminine plural noun sillas.

Could I leave out Nosotros and just start the sentence with Necesitamos?