Familiars can be obtained by multiple means, although it might require some time.
That said, if you mean a familiar as described strictly in the Find Familiar (FF) spell found in the PHB, then the spell is really the only way to get everything described. Other methods bring about their own advantages and disadvantages, which is what is discussed here. For the familiars gained from the spell or from the Warlock's Pact of the Chain, see the PHB for all the rules.
Before listing the different approaches possible, it is worth noting that having a character possess a familiar can drastically alter the mechanic of your party, ranging from exploration, to social encounters, all the way to combat (this will be discussed later).
What exactly is a familiar?
A familiar is a tiny creature which possesses some form of connection to a character (usually a caster). This connection allows for communication between the familiar and its owner, usually telepathic, and often also allows the owner to perceive through the familiar's senses.
Gaining a familiar
Variant rules (these will have different abilities):
Pact: According to certain variant rules in the MM, some creatures (celestials, fey or fiends) can become familiars by forming a contract with them. There are no rules on how much time this takes, but it must be enough to actually summon the creature, negotiate with it and seal the pact. An example of this is the Imp (variant found in MM, pg. 69).
Creation: Some casters have the possibility to create familiars and other allies. The only one specified by RAW is the Homonculous (MM pg. 188). Again, creation time isn't specified in RAW, but it should also take around a year, in my opinion (or use Magic Item creation rules, DMG, pg. 128-129 and attribute it a rarity level).
Familiars gained through these methods are NPCs and therefore are not bound to its "owner" in the same way, usually being controlled by the DM instead of the player, and while they're considered extremely loyal to the PC to which they are bonded, if a pact is broken, or if the familiar is mistreated, it is free to leave or disregard commands. Also, while they do not gain the abilities of familiars listed in the FF spell (such as delivering spells and being stored in pocket dimension), they have access to all of their normal abilities and actions. In the case of familiars gained through pacts, the only feature added is the telepathic bond.
Homebrew rules (playtested in various campaigns):
- Taming: A tiny creature could become a familiar if it is tamed (requires Int of 4 or less). According to downtime rules, this takes bout a year and requires Animal Handling checks (UA).
The main difference with this is that a beast, or a typically non-magical creature, is not able to form the same telepathic bond with its master as the other types of familiars. Also, having an intelligence of less than 4, it follows its master's commands almost blindly, but cannot be issued complex commands (a few words, up to a simple sentence). Any player character can possess such an NPC companion, but casters are able to go further with theirs if the following is allowed:
Casters who have access to the Speak with Animals and the Beast Sense spells can create a telepathic bond with their familiar by cast it on them everyday until the familiar is tamed. This forms a telepathic bond with their familiar, allowing them to issue more complex orders (a few sentences), communicate with them in for a range of up to a mile, and sense what their familiar can sense using an action with the same range.