There are many varieties of ten-string guitar, including:
Harp guitars are guitars to which extra strings have been added which are never fretted but may be plucked or strummed. These strings are therefore played in a manner similar to those of the harp, while those of the principal neck are played as a guitar, hence the name.
Often a second neck, parallel to the fretboard, carries these extra strings. There have been many designs of harp guitar, but in the nineteenth century ten-string versions were particularly popular. Information on nineteenth-century harp guitars comes from three main primary sources:
In the early 19th century Ferdinando Carulli and René Lacôte developed a harp guitar they called the Décacorde (French for "ten-string"). Carulli played this type of guitar and wrote a method for it titled Méthode Complète pour le Décacorde. In it he describes the tuning as C-D-E-F-G-A-d-g-b-e' (strings 10 to 1), with the upper five strings A-d-g-b-e' fretted and the lower basses C-D-E-F-G not fretted. Carulli also wrote divertissements for this instrument.
Two Décacordes by Lacôte are housed in the Music Museum of the Cité de la Musique in Paris:
There is also a Décacorde (attributed to Lacôte) that was in the workshop of Françoise Sinier de Ridder, which has 7 strings on the neck (fretted) and 3 sub-basses (unfretted strings).
Sinier and de Ridder have pointed out that the décacorde was made in three different string configurations. Those instruments that adhere to the Carulli patent have 5 strings on the fingerboard and 5 floating basses [...]. Other specimens that do not bear the patent stamp are known with 6 strings on the fingerboard and 4 floating, and 7 strings on the fingerboard and 3 floating. I now speculate that these latter may have been configured, not as true Carulli Patent Décacordes, but as similar-appearing Lacôte ten-strings tuned more traditionally, and perhaps, played "professionally." 
Period harp guitars built by Johann Gottfried Scherzer survive. A copy of one of these, based on an original circa 1862, has six fretted and four unfretted strings.
The extended-range classical guitar is a classical guitar with additional strings, normally extra bass strings past the bass E string, that are available on the fingerboard.
Many configurations have been produced, but the ten-string classical guitar received a particular boost in 1964, when Narciso Yepes performed the Concierto de Aranjuez with the Berlin Philharmonic Orchestra, using a ten-string guitar invented by Yepes in collaboration with José Ramírez III, with a specific tuning designed to supply sympathetic string resonance to all twelve notes of the chromatic scale, in unison with any note played on the treble strings. This was significant for two reasons:
The use of the ten-string classical guitar is similar to that of the harp guitar:
Unlike the harp guitar, the extended-range classical guitar has a single neck and allows all strings to be fretted.
While the six-string classical guitar remains the standard and most common instrument, since 1963 ten-string guitars in similar configuration to the original Ramírez have been adopted by many classical guitarists and produced by several first-class luthiers, using both Yepes' original tuning and others.
The Halo Guitars XSI is a solid body ten-string guitar with ten individual steel strings, two EMG ten-string pickups, and a Kahler ten-string tremolo bridge. XSI is an acronym for Extended Scale Instrument. Halo Guitars tunes the XSI from low to high as follows: A+D+G+C+F-B?-E?-A?-C-F.
Agile retails a number of extended range guitars, including 10 string instruments in their Interceptor line. Also, two or three times a year they open a custom shop where 10 string guitars can be built with personalized features for the Septor, Interceptor, Intrepid and Pendulum models.
In January, 2009, Gadotti Guitars announced the 10 String Nylon King Electric, a solid body, nylon-stringed ten-string guitar, suitable for both Yepes and other tunings such as the Baroque.
A ten-string jazz guitar by Mike Shishkov, based on the ten-string extended-range classical guitar, was demonstrated at the 3rd International Ten String Guitar Festival in October 2008.
Hawaiian guitars are electric lap steel and table steel guitars with six, eight or ten strings per neck, and one or two necks. The ten-string single-neck instrument is one of the standard configurations, not one of the most common but not unusual either.
Most pedal steel guitars have either one or two ten-string necks. Some but by no means all advanced players use necks with more than ten strings, but ten strings is the normal minimum.
The standard student pedal steel guitar is a single-neck ten-string instrument with three pedals and from one to five knee levers, tuned to E9 tuning.
The first step up from this is a professional S-10 with three or more pedals and four or five knee levers, and the most common next step up is to a D-10 with eight pedals and five knee levers. The D-10 is the most common configuration for professional players.
Some advanced players prefer to remain on an S-10 configuration, perhaps adding more pedals and/or knee levers. Other advanced players progress from the S-10 to a single neck instrument with twelve strings, either a U-12 which uses a universal tuning, or an S-12 which uses an extended E-9 tuning. Single neck instruments with more than twelve strings also exist, such as the fifteen-string universal tuning U-15, and double-neck with more than ten strings per neck, notably the D-12 with two twelve-string necks and various tunings most commonly based on extended E9 and extended C6 tunings.
Professional instruments are normally custom-made to order. Even in the case of an S-10, while the first three pedals and five knee levers are fairly standard in function, there are variations to the order of these and many players add others. Advanced players of all configurations tend to design their own individual setups, known as copedents, specifying the exact string tunings and gauges and the actions of the pedals and levers.
The baroque guitar is one of the earliest instruments considered a guitar, and the first to have significant surviving repertoire.
Surviving baroque guitars have (or originally had) nine or ten strings, in five courses.Stradivarius guitars (of which two, the Hill (1688) and Rawlins (1700) survive complete, plus a neck and several other fragments) all had ten strings in five courses.
The English guitar is a type of cittern that was particularly popular in Europe from around 1750-1850. The English guitar has a pear-shaped body, a flat base, and a short neck. Its distinguishing feature is that it has ten strings in six courses, of which the highest eight are paired in four courses (duplicated strings) with the two lowest strings in two separate courses. This is the same stringing as was later used for the B.C. Rich Bich 10 Guitar, although the traditional tuning for the English guitar is a repetitive open C tuning (C E GG cc ee gg).
The viola guitar is a guitar with ten light steel strings in five courses, played with the fingers rather than with a plectrum. It is particularly prevalent in the folk music of Brazil, where it's called "viola caipira" (country guitar) or simply "viola." The viola braguesa and viola amarantina are other types of ten-string Portuguese folk guitars, which are possibly predecessors of the Brazilian instrument.
These are six-course instruments, unlike most ten-string guitars which either have ten individual strings or five two-string courses. The B.C. Rich ten-string is tuned and played similarly to a six-string but with two-string courses in place of four of the single strings of the six-string.
This instrument was introduced as a custom order model with a new body shape known as the Rich Bich at the 1978 NAMM Show. There were two innovative features:
The B.C. Rich "Bich" 10-string guitar was developed by Neal Moser. During the late 1960s, he was a technical consultant for Jimi Hendrix, and Steven Stills of Crosby, Stills, and Nash. He worked as a sub-contractor for Bernie Rico (B.C. Rich) from 1974 to 1985. During his time with B.C. Rich, he conceived, designed, and built the first Bich 10-string prototype guitar.
Contrary to popular belief, the Bich guitar was never owned by B.C. Rich. The design was licensed to B.C. Rich under contract with Neal Moser. A lawsuit between Neal Moser and HHI Holdings Inc./B.C. Rich was settled. This settlement gave Moser Custom Guitars and HHI/B.C. Rich the right to produce their own version of the Bich 10-string and 6-string guitars. In 2003 B.C. Rich contacted Neal Moser, Sal Gonzales, and Mal Stitch to produce 25 reissue models of the original prototype Bich 10-string guitar. Due to the filing of the lawsuit, only 15 B.C. Rich/PMS 25th Anniversary Prototype Bich 10 string guitars would ever be crafted.
The design was successful enough to be still in production as a ten-string, but many players also bought it for the body shape rather than the ten-string feature and simply removed the extra strings. B.C. Rich recognized this by releasing six-string models of the Bich body shape.
Close relatives of the guitar with ten strings include: