One of my players has a Warlock character and is using Pact of the Chain for their Pact Boon. They most often cast a pseudodragon as their familiar, but neither of us are sure about how they should be using the pseudodragon during fights. Does the familiar get its own turn in the turn order? Or can the familiar only act when the Warlock gives up their action slot? I read the PHB content, but I'm still unclear how/if the pseudodragon should be used in fights.
Your options are found in two different places: find familiar and the Pact of the Chain feature description.
We begin with find familiar:
Your familiar acts independently of you, but it always obeys your commands. In combat, it rolls its own initiative and acts on its own turn. A familiar can't attack, but it can take other actions as normal. [...]
Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. If the spell requires an attack roll, you use your attack modifier for the roll.
Normally, familiars sumoned by find familiar cannot attack. Pact of the Chain changes this:
Additionally, when you take the Attack action, you can forgo one of your own attacks to allow your familiar to make one attack with its reaction.
Whether or not you should do this is the subject of some discussion on this Q&A: For a Pact of the Chain warlock, is choosing to have their familiar attack generally the worse option for the warlock's action?
Notably, clever use of the Help action is usually the best thing a familiar can do in combat: Can a familiar use a Help action to grant advantage on an attack to a near by ally?
Breaking this down, here's a bulleted list of how this works:
- The familiar rolls its own initiative (1d20 plus dex modifier) and takes its own turn.
- The familiar cannot attack on its turn.
- The familiar can take other actions in combat, such as Dodge, Dash, Help, Disengage, Hide, Search, Ready, Use an Object.
- On your turn, you can forgo one of your attacks to have you familiar attack, expending the familiar's reaction (this feature is unique to Pact of the Chain).
- On your turn, when you cast a spell with a range of touch, the familiar can deliver the spell as if it had cast it.
Certain Eldritch Invocations can improve the familiar.
Most notably, the invocation Investment of the Chain Master from Tasha's Cauldron of Everything significantly improves the familiar:
- The familiar gains either a flying speed or a swimming speed (your choice) of 40 feet.
- As a bonus action, you can command the familiar to take the Attack action.
- The familiar’s weapon attacks are considered magical for the purpose of overcoming immunity and resistance to nonmagical attacks.
- If the familiar forces a creature to make a saving throw, it uses your spell save DC.
- When the familiar takes damage, you can use your reaction to grant it resistance against that damage.
With this feature, when using your bonus action to command the familiar to attack, it is somewhat unclear exactly when the familiar takes the attack action, see this Q&A where I argue that it waits for its turn before taking the attack action: When does a warlock's familiar make their attack if the warlock has the Investment of the Chain Master eldritch invocation?
There are several more invocations that interact with the Pact of the Chain familiar, they can be found here on DNDBeyond.