The name "capo" comes from the Italian capo tasto, meaning head fret; the term was originally applied to the nut of the guitar. The name came to be applied to a device which essentially changed where the "head fret" was by resting on the fretboard and fretting the strings for the player, changing the tuning without any other effort.
The first known capo was a flattened-C-shaped piece of brass, used in the 1700s. It was forced over the instrument's neck and held in place by the metal's own tension. It was probably not amazingly effective, and the seemingly unpadded metal probably did quite a bit of damage to the guitar.
In the late 1700s, two other capos came into use, the English yoke and the Spanish cejilla. The former was shaped, unsurprisingly, like a yoke, and had a crossbar padded with leather that rested on the strings. This capo was secured by means of a screw going through the middle of the yoke, usually pressing a pad against the neck of the guitar, but at least on some English guitars, the screw actually penetrated the neck. The cejilla gained the name "little eyebrow" from its shape, or possibly from its string. The capo is a large wooden block with a screw at the top; a string is threaded through a hole in the screw, looped around the guitar, and tied to the other side, and the device is fastened to the guitar by turning the screw, drawing the string tighter.
Capo designs kept improving through the years. In 1850, James Ashborn applied for the first American patent of a capo, similar in design to the yoke capo, but tightened by means of a lever with a cam on it and held in place by its own pressure. In the 1890s, a D-shaped capo was sold by the Gutman company, tightening by a similar mechanism but with the lever on the top instead.
Sears sold a strange guitar in the early 20th century, having a built-in capo. The capo was tightened by means of a string, and a slot was cut in the neck to allow the capo to slide up and down the fretboard.
The elastic capo was patented in 1931 by W.H. Russel. This is one of the simplest capos, being a hard, padded bar held fast to the guitar by means of an elastic strap winding around the neck.
James Dunlop, the Jim Dunlop of Jim Dunlop, patented a "toggle" capo in 1965; similar to the cejilla, this device worked by inserting a lever into one of several holes in the top of the capo and folding the lever down, tightening the strap against the back of the neck.
The first plastic capo was patented in Germany by Herbert Bauerfeird in 1973. This device was a broken loop, with a pad placed over a set of reeds cut into the top of the loop in order to accommodate the different curves and tensions that various guitars provide.
R. Shubb redesigned the side-clamp capo in 1978, adding an adjustable screw to the bottom lever, allowing the user to adjust the lever to accomodate the width of the guitar's neck. This capo is now popularly referred to by the creator's name.
"Trigger" capos, held in place by a spring on the side, were patented in 1979 by Nichols, Berner, and Fernandez, with inspiration provided by the clothespins Berner's wife used. The various shapes were licensed to various manufacturers; one to Jim Dunlop (now the possessor of the Trigger name), and several others to Kyser. Some Kyser models were even designed with a device in the lever to assist in pulling bridge pins.
In 1986, Swany Cornette patented the "Glider" capo, based on earlier rolling capos; this capo has rollers on the top and bottom, allowing for quick key changes.
One of the oddest capo designs was the Third Hand Capo. Similar in design to the elastic capo, but the bar is cut into six pieces. The pieces can be rotated around the bar so that only certain strings are fretted. This allows the guitar to have some of the advantages of alternate tunings without having to retune the guitar. Lyle Shabram, Jr. patented this in 1980.
WHY YOU WOULD USE A CAPO
The capo raises the pitch of the open strings. This allows guitarists to change the key of a song (usually to accomodate the vocal range of the singer) without having to change the way they finger the chords. Using a capo, the guitarist could play a standard C chord shape and have it sound as a D, or E, or F# without changing the shape of his/her fingers by placing the capo on the second, fourth, or sixth fret and fretting the chord as if the third (or whatever) fret was the first fret; for all intents and purposes, the third fret IS now the first fret.
The change in placement also changes the sound of the strings; flamenco guitarists often use a capo to brighten their sound.
Also, when using alternate tunings, it is often better for the neck to tune the guitar down and then use the capo to raise the pitch.
CAPO ADVANTAGES AND DISADVANTAGES
When you capo your guitar, you bend the strings. This can throw them out of pitch. This can be rectified by placing the capo as close to the fret as possible. The strings can also be bent sideways during application, requiring readjustment of the capo.
The capo itself can also interfere with the guitarist's playing hand; the device obviously sits higher on the fretboard than anything else, so the guitarist may bump his/her hand on the capo, interfering with his/her playing and possibly knocking the capo out of alignment.
Advantages and disadvantages vary from capo to capo, as well. These are the ones i have used, and my takes on them. Your mileage, of course, may vary.
SHUBB: Low profile (that is, small), so that the fret doesn't get in the way of the guitarist's hand, and fairly easy to change; however, these must be stored elsewhere when not in use, are usually fairly expensive.
TRIGGER: Quick and extremely easy to change, can be clamped on headstock when not in use; but it may bend strings sideways on application, can have a high profile, and somewhat expensive.
ELASTIC: Cheap, fairly easy to apply, very light and easy to carry around; not amazingly secure or reliable, can be replaced with a pencil held in place by a rubber band with similar results.
TOGGLE: Fairly inexpensive, fairly reliable, very small; can be a pain to adjust, and somewhat difficult to change.