Fingerpicking is a method of setting the guitar strings in motion using the fingers, rather than a pick or a toothbrush, etc. It's like having 4 picks rather than one.
Fingerstyle is a term meaning fingerpicking, but typically on a steel string guitar. We can fingerpick (play fingerstyle) on any type of guitar, yet the term is closely associated with steel string guitars. Jazz players also use the term for finger comping too. Fingerpicking = Fingerstyle.
It is important to have the most natural mechanics - this creates great tone - round & resonant. Using our hands in alliance with nature & human physiology gets the best results.
To get great tone, it all starts with the drivers and the main knuckle joints for the fingers are the primary drivers. By using this knuckle, the finger will rebound. The thumb's driver is the joint at the wrist.
The fingers pluck upward into the hand and the thumb plucks downward (they both strum in both directions). For hand symbols, check out this lesson.

As far as using the main knuckle, think about how we pick things up. Can we effectively pick up anything effectively with the upper finger joints?
Again, the main driver for fingerpicking is the joint at the hand. This offers us rebound. When we pluck with this joint, the finger will return to its position. If we are getting ready to pluck a different string, we move your arm to get a different fingertip location.
When we use the main knuckle driver, it automatically snaps back to where it was before [this is called rebound]. Rebound is automatic and is built into our hands [when using the primary driver = the other knuckles have a slower, less reliable recoil].
This is how we track where our fingertips are located. We pluck and forget about the rebound [a single blip of thought completes a whole stroke cycle]. It happens naturally. To pick different strings, we move our arm, which moves our hand. We keep the motion consistent for any string. Same stroke, different arm position.
A goal in learning to fingerpick is to discover how our hands work and what type of motion gets great tone.
In modern playing, we play thumb out. The thumb plucks to the outside of the hand, as when you make a fist.

The thumb & fingers need a pathway; a path that doesn't interfere with the follow through of the others.
Always keep in mind: thumb plucks down, outside the fingers, & the fingers pluck up, inside the thumb.
We let our fingers rebound - come back to where they started. This allows our physiology to do most of the work. We naturally know where our fingertip is located.
Let's get our motors moving. We will play some arpeggio patterns, a finger flutter exercise, & some blocks (playing tones together as a block).
Let's start with an arpeggio [broken chord] exercise. See hand symbols if needed. p = thumb, i = index, m = middle, a = 'ring', and c = pinky.
Pick any chord or use open strings

Use all six combinations of ima, sticking with the string sets: ima as shown above, iam, mia, mai, aim, ami. So this would be 6 exercises. We can also repeat each measure above as many times as needed to get full control. Maybe 2 or 4 times each.
Now, let's stick with those string sets, but mix up the patterns (again, we use all six combinations of ima: ima as shown above, iam, mia, mai, aim, ami). So this would be 6 exercises. And again, repeat each measure above as many times as needed to get total control. Maybe 2 or 4 times each.

We can use any chord that utilizes 6 strings, such as E or Em, or just use open strings.
Try to keep the driver knuckles over the string set we are picking. The space between your thumb & index will open up, & our elbow will move our hand over the next string set.
Now, let's try an arpeggio pattern that is popular for classical guitar, & a great one to know...

Next, let's do some finger flutters. When we train, we should strengthen the muscles of your arms & hands in both directions (extension/flexion). When we pluck or pick towards the hand, we are flexing the muscle. When we do rasgueado [to scratch or brush - finger strum], we are extending.

An up arrow is a down strum; a down arrow is an up strum [consistent with the tablature]. Explore your full range of motion for each finger individually.
And finally, let's get all the fingers moving together in blocks.

When we train, we alternate between arpeggios, finger flutters, & blocks. This is cross-training. Study your motion. Get comfortable. Find your picking pocket.