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.
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.
For wizards there are three ways to obtain a familiar: (1) via the find familiar spell, (2) by adventuring and meeting a creature that itself has the ability to become a familiar and is willing to do so, (3) by meeting a creature that they can bond with as their familiar by casting find familiar on it.
All but the first one require your DM's buy-in to create more colorful wizard characters. The main concern there is that using other creatures may be a power play or unbalancing, so be prepared for the DM to say no to more exotic options for your PC. Having a tree-hugging Elf Wizard with a Pixie familiar may be flavorful, but also would be entirely overpowered at low levels.
In this case the familiar is a fey, celestial or fiend spirit that takes one of the beast forms described in the spell, and has the abilities described in the spell.
In addition to the beasts in the spell, various other creatures have been listed in official adventures as options for summoning with the spell if the DM agrees to it: a Tressym, Almiraj, Flying Monkey, Abyssal Chicken, Fox, and Hare.
Strixhaven: A Curriculum of Chaos has a feat called Strixhaven Mascot (preconditions are 4th level and another feat, Strixhaven Initiate), that allows you to chose the following option as your familiar, depending on your college: spirit statue mascot, art elemental mascot, fractal mascot, inkling mascot or pest mascot.
While these are the ones listed in published materials, talk to your DM if you want another familiar. If it is just for flavor, reskinning the statistics of an existing one should be unproblematic (a toad with frog stats, a hamster with weasel stats etc).
As mentioned in the NPC section in the MM (p. 374) and in Volo's Guide to Monsters (p. 213), for NPC spellcasters that can cast find familiar any Tiny monster might be an option, such as a Crawling Claw or Cranium Rat.
The Monster Manual lists a number of creatures that can act as familiars. These monsters include: the pseudodragon, quasit, and imp. Volo's Guide to Monsters adds the gazer (which requires a 3rd level or higher spellcaster to bond with).
These monsters serve as familiars of their own accord, and can end the relationship at any point if they so chose. This means it is up to the DM to decide if one of them will be your familiar. They are not limited to wizards as companions.
They differ from the familiars summoned by find familiar, in that they are actual creatures that will die (or be sent back to their home plane) when killed. Find familiar does not impose any limitations on them or grant you any abilities for them. For example, they can attack in combat and you cannot send them to a pocket dimension or deliver touch spells with them. The bonding abilities you enjoy when they are your familiar are granted by them, not the spell. They include:
a telepathic bond with its willing master. While the two are bonded, the master can sense what the familiar senses as long as they are within 1 mile of each other
while they are within 10 feet of their companion, the companion shares their Magic Resistance trait [for pseudodragon, quasit, and imp]
This is a mix of the other two options. The familiar in this case is a creature encountered adventuring, but the adventure requires the wizard to cast find familiar to bond with them.
In both cases the adventure text does not elaborate in detail how the bond works once established, and this will be up to the DM.
In our case, the DM ruled for the gazer that it behaves as described above, the range for the telepathic bond is 1 mile, none of the abilities or restrictions of find familiar apply, except that it has no freedom to leave the bond. (Our DM also allows casting find familiar on other Tiny creatures to bond them, but reserves the right to veto -- no Fairy Dragon familiar, for example.)
