Bonesingers are Eldar artisans, weapon crafters, architects, masons, IT technicians, janitors, coffee brewers, firefighters, yadda yadda yadda. They possess the unique ability to shape the Wraithbone, without which the Eldar would be armless failures.
Essentially speaking, Bonesingers are the Eldar equivalent to the Imperium's Enginseer, but fundamentally less awesome and more psychic. This is relatively true in both the fluff and crunch and they behave almost like carbon cutouts of each other.
These Eldar follow one of the many Paths along which their kind journey. Rather than focusing on combat, the Bonesinger is instead tasked with the construction and maintenance of Eldar structures and machinery. Usually through the psychic manipulation of Wraithbone, shaping and molding it into various items. This is done true psychically molding the fabric of Wraithbone through a 'chorus' of psychic energy, hence why they are called Bonesingers; they literally make structures through the power of music. You can't get more Elvish than that. Bonesingers of the Il-Kaithe Craftworld are said to be the most skilled of the Eldar Bonesingers, able to practice their art even in the heat of battle.
While Wraithbone is capable of naturally repairing itself over time, the process can be greatly enhanced by the psychic abilities of a Bonesinger. Hence, whilst they practically never see combat, unless a Wraithlord breaks a leg or a Falcon's weapons go derp, they are always in the backline. Still, they managed to be in Dawn of War as the Eldar builder unit.
Curiously, while Fluff states that Bonesingers are not limited by gender, all of the Bonesingers in the Dawn of War games are women (or at least they sound like women, cause god knows how anybody is gonna recognize the gender of Eldar synth-voices). As you can imagine, with the name 'Bonesinger' and them being all women, it has led some to believe that they may also perform another use for....'Bonesinging'...
On the Tabletop
Thanks to Made to Order, the old, sexy Bonesinger model was re-released for just one week only. He was also released with free rules, which you can check on the Warhammer Community page.
In practice, the Bonesinger functions similarly to the Imperium's enginseer, able to heal any Wraith Construct or Vehicle for D3 wounds a turn. Unlike the Enginseer, they do this during the Psychic phase and must sacrifice their smite cast in order to do so. Given the limited range of units they can heal, and given that they cannot resurrect slain units like the Space Marine Apothecary can, they are objectively less efficient as a support unit compared to many other army counterparts. However, as they can still function as a vector for mortal wounds through smite and can still attempt to deny opponent casts all the same, they can make up for their lack of healing support through their heightened offensive presence.