Nothing.
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.
Firstly, yes, if a spell requires an attack, that attack still counts as an attack.
So, what you have to remember is the "specific beats general" rule. This is detailed on page 7 of the PHB. The core of the rule is:
If a specific rule contradicts a general rule, the specific rule wins.
So the general rule here is that familiars can't attack. There are 2 exceptions to this general rule:
A familiar can deliver spells with a range of touch on your behalf, even if the spell requires an attack roll.
A Warlock with the Pact of the Chain can forgo one of their attacks to let their familiar make one.
So when you cast a spell with a range of touch, you can have your familiar do the actual "touching" on your behalf. You cast the spell as normal using whatever actions it requires. The familiar is required to use their reaction, and if the spell requires an attack roll, it makes the attack roll. Since it uses your modifier anyway, it's exactly the same as you making the attack roll except that you don't have to stand next to what you're casting the spell on.
In the case of a Warlock with the Pact of the Chain, the Warlock takes the Attack action as usual, then has the familiar do the actual attack. The errata for the PHB says:
When you
let your familiar attack, it does so with its
reaction.
If the Warlock can make multiple attacks with the Attack action, the familiar can use its reaction to replace one of them with its own attack, then the Warlock makes the rest.
Finally, only spells with a range of touch can be delivered by your familiar. There is no option for having your familiar cast a spell on your behalf, and the option for allowing a familiar to attack instead of yourself specifically says "when you take the Attack action", not just "when you make an attack".
Best Answer
The unconscious condition says:
and incapacitated says:
So:
No, the wizard casts the spell, the familiar only delivers it.
(emphasis mine)
Since the wizard is incapacitated, they cannot cast spells. Since they cannot cast spells, the familiar cannot deliver them.
The wizard being unconscious does not affect anything the familiar could already do on its own as it is perfectly capable of independent action. That means that a wizard could have arranged for the familiar to follow actions in the case of the wizard getting knocked out.
What actions they are capable of taking is going to depend on the situation, the statistics (especially Intelligence score which is usually between 2 and 4), and the physical characteristics of the familiar. Your DM will have to work with you to decide what a familiar can do and what it actually does. For more discussion on this see What can a familiar actually do?.