In Final Fantasy VII, the TURKS are tools of Shin-Ra, and are basically there to do the company's dirty work. At the time of the game, there are four Turks: Tseng, the leader, Reno, Rude, and Elena, the rookie.
The name Turks is most obviously traced to Turkey, a country in the Middle East.People from Turkey are called Turks. Historically, westerners have viewed the Turks as infidels, and so the archaic definition of Turk is "tyrannical and cruel". Another, non-geograpical definition of turk is "an unusually young dynamic person eager for change," from the Young Turks, an early 20th century revolutionaly party that was largely responsible for the Armenian Genocide. They supported the Central Powers during the Great War, and lost power after that group lost the war. Because Final Fantasy is notorious for its naming symbolism, the name "Turks" probably has its origins in one or both of these definitions.
Rude: Do you know who I am?
Cloud: From the Turks, right?
Rude: Well if you know, then this won't take long. It's difficult to explain what the Turks do...
Cloud: Kidnapping, right?
Rude: To put it negatively... you could say that. ...But, that's not all there is to it, anymore.
The Turks are basically an elite mercenary branch of Shinra. They scout for SOLDIER (Shinra's genetically engineered army), finding possible recruits, and they do other odd jobs that require their special brand of expertise. They are a unique brand of evil in the game- they are emminently human characters and are often used for comedic effect. They share a similar mission with your party- apprehend Sephiroth- and the player team and the Turks even team up for one incedent.
The Turks share a fierce loyalty between themselves. Reno, for example, shows no particular loyalty to Shinra, but obviously takes pride in being a Turk. They even vacation together. (In their uniforms even!) At the end of the game, you can choose whether or not to fight a final battle with the Turks. They realize Shinra's going under, and they don't seem to care- but they do mourn the loss of their orginization. They seem to be pretty close, sharing crushes and teasing the new recruit. They're like old war buddies.
The Turks have been around for at least thirty years before Final Fantasy VII takes place. Vincent, an optional member of your party, is a former Turk, and though he has some skills with guns and a snappy uniform in a flashback, he's not evil. Just sort of moany and depressed. It's Vincent who identifies the TURKS as the "Shin-Ra Manufacturing Department in Administrative Research." But then, he was a Turk three decades ago, and lots about Shin-Ra has changed. It's actually a pretty recent developement. It was the president who was killed in the first act of the game took Shin-Ra from run-of-the mill power conglomerate to military-industrial complex.
During that time, the inital discovery of JENOVA led to the department of Hojo's crazed experiments, and eventually to the creation of SOLDIER, the elite military wing of Shin-Ra. Once the company got guns, it was kind of all downhill.
The primary official capacity of the TURKS is scouting out SOLDIER recruits. They are the press gang. But it's pretty clear that what they actually do is the dirty work.
Six years before Final Fantasy VII, when Before Crisis takes place, the Turks fight a war with a much more organized AVALANCHE (an eco-terrorism group that your main characters are affiliated with.) The catch? In Before Crisis, you play as the Turks. It turns out that the AVALANCHE that came before wasn't exactly all sunshine and rainbows, either. It also becomes pretty clear that they have extreme moral reservations about what they're beginning to be asked to do. But by the time Final Fantasy VII rolls around, they've squared with their consciences and made peace with the nastier aspects of their job descriptions.
In Advent Children, the TURKS are still around, working for Rufus directly as he tries to rebuilt Shin-Ra. The man calling the shots has reformed to some degree, and wants to fix what he broke. The TURKS are no longer recruiting/kidnapping for SOLDIER, but are just doing random odd jobs and acting as Rufus's bodygaurd.
Of course, the best part about being Turk is the thing that's the most constant: the snazzy uniform. Smokin'!