In the class section for wizards in the PHB it doesn't state anything about familiars. You can learn the find familiar spell as a 1st level spell which creates a "spirit that takes an animal form", but can actual creatures or animals be obtained as a familiar? The MM mentions the pseudodragon being a familiar so I would think that it is possible but haven't been able to find any rules on the possibility of doing so or what creatures can be familiars.
The Find Familiar spell is currently the only method for a wizard to obtain a familiar, and it lists the creatures available. However, a large number of creatures in the Monster Manual, such as the pseudodragon, imp, or quasit, include the 'Familiar' variant. Note that a Warlock with the Pact of the Chain feature can obtain many of these creatures as a familiar.
What you have to remember is that variant monsters, like monsters, are designed for the GM to use to make enemies more interesting. The Familiar variant is a monster variant, just like the troll's Loathsome Limbs variant or the Genie Powers variant. They're for GM use rather than player use.
The Mage NPC in Appendix B of the Monster Manual also has a Familiar variant, which says:
Any spellcaster that can cast the find familiar spell (such as an archmage or mage) is likely to have a familiar. The familiar can be one of the creatures described in the spell, (see the Player's Handbook) or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit.
So the Familiar variant is for GMs to create more interesting NPCs, rather than to provide players with additional options. Of course, with your GM's permission, you could obtain a more interesting familiar. This would probably involve actually finding such a creature and somehow forming a bond with it. But this relies solely on your GM to allow and arbitrate.
It's also worth considering that allowing a wizard to obtain one of these more powerful creatures somewhat invalidates the Warlock's Pact of the Chain.
Yes. Some creatures in the Monster Manual have variant familiar rules. These rules do not require the Find Familiar spell. In fact, the variant familiar rules are completely separate from the Find Familiar spell in everything but name. In order to gain a variant familiar, the player must find one of these creatures during gameplay and convince it to serve them. The details of what this entails are left up to the DM.
The extent of a variant familiar bond is detailed entirely in the box on the creature's Monster Manual entry. Some features overlap with those of Find Familiar, while others do not.
To contrast the differences between an ordinary familiar and a variant familiar, I'll use the Pseudodragon. The Pseudodragon as a familiar is not obligated to follow its master's commands, and can choose to leave if it feels mistreated. It also does not have the ability to deliver touch spells for its master, but does have several abilities which Find Familiar does not grant, such as a longer range on the shared senses ability, and shared magic resistance while within 10 feet of its master.
Most importantly, however, the pseudodragon is a mortal creature, unlike a familiar from the Find Familiar spell. When its hit points are reduced to zero, it will not simply be unsummoned, but will either die or begin making death saving throws.