Reading over the PHB spell entry for familiars, I don't see anything about what happens to the familiar if its "owner" dies or is unconscious. I'm going to assume if the owner dies, the familiar disappears, but what if the owner is merely knocked out?
There is no current rule that spell effects end when a caster is unconscious or dies unless they are specifically concentration spells.
Thus, since Find Familiar is not a concentration spell, it's perfectly reasonable to assume that the familiar stays and is unaffected by the fact that his wizard is unconscious (or dead).
And in fact, the duration of the Find Familiar spell is "instantaneous" meaning that once the familiar is created, it's there, and there is no continuing effect of the spell. It can't even be dispelled.
It's important to note that if your DM wants to rule that current spell effects (like say, spiritual weapon), end when you go unconscious, it would not affect a spell like Find Familiar since again, the duration of that spell is instantaneous.
Technically, Nothing. (Ok, maybe, if the Wizard dies, then, possibly it 'disappears' from the story...or self-suicides)
Ok..reading the 5e rules,, it's still a 'spirit' forced into a shape of a common animal...which, doesn't necessarily invalidate the point that it can be trained (without necessarily having to take Animal Training...and some bonus could be given due to the fact that you have Telepathic communication)
If the Wizard is unconscious, the familiar would act as an independent creature -- and, according to it's nature/training (e.g. a dog might stay and protect him/her, while a cat might slink away with an eye toward an ambush, or a rat might scurry away)[with the only 5e addition being, it can't take an Attack Action] -- it doesn't proscribe it from executing a Trained action (e.g. The question of could a cat familiar drink a potion). A smart Wizard might create a familiar-triggered magic item that a familiar could use...
The way I've always seen it is a familiar is a permanent summoned creature who is common to your locale who matches your alignment (i.e. it can't be dispelled or banished -- except in 5e it's also essentially immortal)