The class section for wizards in the PHB 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 that the pseudodragon can be 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.
Best Answer
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:
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.