The term fingerstyle does not refer to a specific style or genre of music, but rather to a technique that involves playing the guitar by plucking the strings with the fingernails or fingertips, as opposed to doing it with a plectrum or pick. The universe of possibilities that the fingerstyle technique creates makes it look more like a whole different approach to the instrument than a particular technique.
Fingerstyle arguably makes the most of the polyphonic capabilities of the guitar. As strings are struck by individual digits, guitarists are able to play more than one melodic line, and thus cover several textural elements as once.
It is possible to extract a great variety of sounds from the fingerstyle technique, depending on factors such as the length and shape of nails, the possibility to use the fingertips instead of nails, the point where the fingers make contact wit the string, the angle at which they make contact, etc. There is also less need to develop a damping technique on the left hand when playing non adjacent strings, as the right hand can avoid the strings in between.The fingers employed individually can attack non-adjacent strings as opposed to the hand used as a single element in traditional picking techniques. Chords become easier to play, whether it be in arpeggiated form (where fingers are a more reliable choice than string skipping) or in block form (up to five strings can be played simultaneously). Complex strumming patterns arise from the individual use of fingers (as in Spanish flamenco).
Conversely, tremolo techniques are much harder to perfect when playing fingerstyle, and it is not as easy to develop speed (starting with the fact that fingers normally attack the strings in a single direction whereas plectrums can alternate directions). Fingernails have to be taken care of, and this usually becomes an enormous issue for fingerstyle playing: they have to be filed, shaped, kept at the right length and thickness. Nylon strings and light gauge steel strings are normally easier to play than heavy gauge strings. Some players compensate for this employing picks that conveniently attach to every finger of the right hand, though, needless to say, these do not yield the same results in terms of tone control.
Fingerstyle is the default technique for when approaching the guitar in a classical/academical fashion. In classical guitar tabs are rarely used (except for beginners instruction, or renaissance and medieval music), with traditional notation being the standard. Fingerings are usually given in the score, either by the publisher or the composer. The left (fretting) hand fingers are referred to as 1, 2, 3, and 4 (1 being the index). The right hand fingers are p (thumb), i (index), m (middle), a (ring), and x (or e, little). Naturally, there are certain conventions in guitar fingerings to be studied.
Even though finger playing is the traditional technique in classical music, some non-academical genres tend to see it as an alternative way of playing. It is often seen in acoustic guitar players, and less frequently in electric guitar players (with notable exceptions being Chet Atkins and Jeff Beck, among others). Flamenco guitar is closely related to the classical approach, though it has its own set of techniques. There is a also distinct set of techniques known as fingerpicking, used in certain forms of folk and country music.
Most complete players will have some notions of fingerstyle, even if they regularly incline to play with a plectrum. The possibilities that arise from playing with the fingers take the guitar to a higher level as a soloist instrument. In this line, it is highly recommended that students develop fingerstyle skills.