Mr. Tseng is the leader of the Turks, at least when they're in the field doing their job. Of course, he answers directly to Heidigger, who then answers to Rufus. We first meet the Turks when they're trying to kidnap Aeris at the beginning of the game. It turns out that Mr. Tseng and Aeris have known each other for some time. Even though it's his job to stop Cloud and Co., he has a soft spot for Aeris. He never makes his feelings known to her, however. You never fight him, but he does have a little pet for you to play with before you get on the submarine.
Elena is the only female member of the Turks and she's also the newest member. She happens to be in love with Mr. Tseng, which is an obvious problem because he's her boss and his heart is too locked on Aeris. You only ever fight her once, and even then you have the option not to. But it is advised. When you return to Midgar at the end of Disc 2 to stop Hojo, you'll run into The Turks in the underground tunnels. You'll be fighting Elena, Rude, and Reno. Elene, herself isn't terribly difficult. Just remember to steal the Minerva Band from her.
His name pretty much says it all. There's not much to say about him other than the fact that when you fight him (and you fight him 4 times). The first time you fight him, he's with Reno on the path to Gongaga. The second time is on the Rocket Platform when you're trying to get the Huge Materia from the rocket. The third is when you fight him with Reno again in the Downed Shinra Plane. The last is when you have to face Rude, Reno, and Elena at once. He's not hard any of those times, but during the last 2 times you fight him, you can steal the Ziedrich armor from him. It has incredibly high defense, but no materia slots.
Reno's not much different from Rude. Both get really drunk and don't feel like fighting when they're on vacation. Anyway, you fight him 4 times, just like Rude, but strategies for him for a little different. Reno is the first Turk you actually fight. After you get out of the Sector 5 Reactor and just before Aeris is kidnapped, you fight him. Be VERY careful when fighting him here. He can encase your characters in trangles and if they're all encased, game over. So, just hit the triangles with your weapons to make them go away. Next, you fight him along with Rude on the path to Gongaga. Watch out because Reno has a gun that will confuse you. Make sure someone has a Peace Ring. You fight him again with Rude in the Downed Shinra Plane. Once again, he uses the confusion gun, but this time you should have 2 Ribbons and a Peace Ring. And you can steal a Tough Ring from him, which is one of the best accessories in the game. The last time you fight him is in the underground tunnels on the way to smack Hojo upside the head. You'll also be facing Rude and Elena and he's got that confusion gun again, which can make fighting the 3 of them a chore, unless you've got your Ribbons and/or Peace Rings. And you can steal a Tough Ring from him here, too.