Do familiars advance in the same way player characters advance? Can they advance at all?
Familiars do not gain class levels, nor do they advance via the Monster Advancement rules from the Bestiary.
Instead, Familiars advance with the level and/or class level of it's master, and follow their own rules for that:
Hit Dice: [... M]aster's character level or the familiar's normal HD total, whichever is higher.
Hit Points: [... H]alf the master's total hit points [...], rounded down [...]
Attacks: [...M]aster's base attack bonus [...] from all his classes. [...]
Saving Throws: [...] Fortitude +2, Reflex +2, Will +0 [...] or the master's [...], whichever is better (for each saving throw). The familiar uses its own ability modifiers [...].
Skills: For each skill in which either the master or the familiar has ranks, use either the normal skill ranks for an animal of that type or the master's skill ranks, whichever is better. In either case, the familiar uses its own ability modifiers. [...]
Furthermore, there are several abilities that depend on the class level of the master, such as its natural armor bonus and Intelligence score. This is detailed in the table following the rules-snippet above, on the same site.
Improved familiars are no exception to the above rules, except
if the creature's type is something other than animal, its type does not change; and improved familiars do not gain the ability to speak with other creatures of their kind (although many of them already have the ability to communicate).
Note that the familiar's intelligence may be lower than usual for it's type based on your class level! A Wizard 7 with Improved Familiar has a dumb specimen for a Nosoi familiar (Int 9 rather than the usual 11 for Nosois).