Pull-ups are the bread and butter of bodyweight training - targeting a large range of muscles from the massive latissimus dorsi to the oft-undertrained forearm flexor muscle group.
Here are popular grips/variations that can target various musculature:
Adopting an underhand or supinated grip transforms the conventional pull-up into the more biceps-focused chin-up.

Though the chin-up replicates the pull-up in practically every aspect, the closer and more supinated grip reduces emphasis on the latissimus dorsi. Instead, the biceps, mid-back and pectorals (to some small degree) are targeted to a greater extent.
Chin-ups will require either a straight pull-up bar or a pair of gymnastic rings positioned overhead.
Like most other bodyweight exercises, programming the chin-up is highly individual and will depend on your current physical capabilities.
However, knowing this, the 3-5 sets of 6-12 repetitions should be a good baseline for the majority of novice and intermediate exercisers.
It’s not uncommon for lifters performing chin-ups to neglect proper scapular engagement in exchange for a focus on elbow flexion. This is a mistake, and can make each repetition significantly more difficult than need be.
In order to perform a chin-up correctly, lifters should ensure that their shoulder blades are retracting as they pull their trunk upwards.
Just as how a close supinated grip targets the biceps to a greater degree, so too does a wide pronated grip achieve the opposite effect - greater latissimus dorsi emphasis.

Though all forms of pull-up do target the lats to some degree, wide grip pull-ups take things to the next level by greatly reducing the involvement of the biceps, trapezius and other secondary muscle groups.
A sufficiently long enough pull-up bar will be needed to accommodate the wider than usual grip used.
Depending on how much of your pull-up technique relies on your secondary mover muscles, wide grip pull-ups can be either easier or harder than conventional pull-ups.
Whether you’re a novice or testing whether a wide grip is right for you, starting with 3-5 sets of 6-10 repetitions is best.
Though the grip used is indeed meant to be significantly wider than shoulder-width apart, avoid setting the hands so far apart that elbow flexion is minimized. Apart from being entirely inefficient, an exceedingly wide grip width will also greatly increase strain placed on the shoulder joint and rotator cuffs.
Whether the lower and middle trapezius, the rhomboids, the teres muscles or the infraspinatus - selecting a narrower pronated grip when doing pull-ups is the way to go.

Like the latissimus dorsi, each of these individual muscles is targeted regardless of what variant of pull-up is being performed. However, setting the hands closer together helps the lifter properly engage them to a greater degree while simultaneously shifting the distribution of resistance away from the latissimus dorsi.
Close grip pull-ups will require a straight pull-up bar - preferably in the form of two separate handles, as this will allow the trunk to move along a more vertical path.
Close grip and conventional pull-ups are approximately the same in terms of intensity and fatigue accrual.
A direct 1:1 transfer of programming is completely feasible - but opting for 2-4 sets of 6-12 repetitions may be adopted instead.
Internal rotation of the shoulder or hunching of the upper back are particularly common mistakes sought with close grip pull-ups. To avoid these issues, ensure the scapula is never protracted at any point of the repetition, and that the chest is “puffed out” throughout the set.
Neutral grip pull-ups are excellent for creating a greater focus on the brachialis, brachioradialis and the upper section of the trapezius muscles.

It achieves this shift in emphasis by having the palms face one another as the hands are set within shoulder-width distance. Like close grip pull-ups, the neutral grip variant reduces the involvement of the latissimus dorsi but is otherwise mechanically identical.
Neutral grip pull-ups will require a pair of parallel bars, rather than a straight pull-up bar.
Depending on how arm-dominant your physiology is, neutral grip pull-ups may be easier than conventional pull-ups.
As such, 3-5 sets of 8-16 repetitions should be a sufficiently intense starting range for novices and intermediates.
As is the case in both close grip pull-ups and chin-ups, ensure that the elbows do not get in the way of the trunk so as to maximize range of motion.
Draw the elbows out to the sides after performing the initial pull through the back - though care must be taken not to flare them immediately outwards, as this can impinge the shoulders.
Unless performing pull-ups as a secondary compound movement, you should specifically select a grip that emphasizes the weakest muscles in your upper body.
For weak arms, chin-ups, or a neutral grip are superior. For the lats or mid-back, a wide grip or close grip are the better choice, respectively.
References
1. Leslie, Kelly & Comfort, Paul. (2013). The Effect of Grip Width and Hand Orientation on Muscle Activity During Pull-ups and the Lat Pull-down. Strength and Conditioning Journal. 35. 75-78. 10.1519/SSC.0b013e318282120e.
