Traditionally, there are three ways: Plectrum (pick), fingers, or combination thereof. (One can, with electric guitars especially, also slap the guitar with various things, including one's hand, to product tones, and even use the fretting hand alone to hammer the strings.)
If you are a dexter, the right hand does this plucking and strumming part, while the left hand presses the strings to the board to form notes or chords.
Picks usually get divided into flat and finger picks, and these are easily recognizable. The flat pick has a plethora of designs, but the basic model is a rounded triangular thing, maybe a credit card thickness or less, and one plucks the strings or strums them sequentially. Most of the rock and blues solos you'll see are done with flat picks, as are most bluegrass, jazz, and folk music.
There are folks who use a pick with two fingers and the other fingers on that hand plucking the strings, a hybrid style.
Finger picks, usually metal, slip over your fingers to produce metal or sometimes plastic fingernails, with the thumb version usually jutting the pick itself out at an angle from the side toward the string. One can strum or pluck.
Fingertips come from classical and flaminco music and instruments, with the more modern "fingerstyle" playing arriving with steel-string acoustic guitars.
Bare fingertips fall into several categories: Organic nails, artificial nails, and no nails.
In classical pedagogy, there is a long-standing argument as to which sounds better, nails or not, and the majority of players opt for fingernails. The actual playing most often involves the nails and fleshy pads in combination.
Most classical or flamingo players opt for their own fingernails. Due to the abrasiveness of steel-string guitars, a lot of players there drop round the Vietnamese nail parlor and have acrylic nails applied. Playing flaminco also involves using the back of the nails, rasqueado, which is hard on them.
Classical teaching is fairly rigorous with which fingers do what on which string. Each digit is assigned a name, from the Spanish, where the classical guitar was mostly designed, i.e., Torres-style instruments. (Later, the Germans did some improvements, ala Hauser-style.)
P-I-M-A, for "Pulgar (thumb), Index, Middle, Annular (ring). The pinky is seldom utilized on the non-chording hand. There are rules as to which finger plucks which string, cross-picking, free-strokes, rest-strokes, angle of attack, la da da da dah.
I won't wander off into that realm, I'd never find my way back. Suffice it to say that fingernails are important to most of those who play classical guitar, and if you break a nail, it can be anything from annoying to a Big Fucking Deal.
If you are about to step on stage to play for a couple thousand people, this falls into the BFD category.
There are all manner of repairs, ping-pong ball pieces glued on used to be high on the list, and there are temporary nails that can be applied with those rubbery glue spots in an emergency.
What I've found that works well for a broken-but-not-detached nail is simple. A thin layer of super-glue, upon which a flat bit of Kleenex is laid, then another drop of glue, and smooth it out with a toothpick. This will last anywhere from several days to more than a week, and can allow you to play normally until the nail grows long enough to file the broken part off.
As you can see from the image, this happened to my middle fingernail, and here is the repair. This is visible if you are looking, but not so awful people will stop and ask, OMG, what happened to your finger?!
Tim Brookes, in his book Guitar: An American Life, quotes a player saying something to the effect of, Yeah, we got into playing guitar to get girls, and now here we are middle-aged men talking to each other about our fingernails ...