Red-eared sliders are very common turtles both in nature and in captivity; they live primarily in the water, but because they are cold-blooded, they do emerge to sunbathe in order to modulate their temperature. They live in diverse habitats--from mud holes with just few hundred gallons to ponds, streams, lakes and large rivers. Red-eared sliders are native to the southern United States, though they are prevalent around the world because they are so popular as pets.
In fact, this turtle is thought to have the highest volume of reproduction worldwide and they are often sold to people as hatchlings. The are probably dumped into the wild both in and out of their native habitat more any other turtle species on the planet.
The fluttering claw movements that red eared sliders sometimes exhibit is most often a courtship ritual or "mating dance." It is most often males that exhibit this behavior, but not exclusively, so you can't use this behavior to distinguish between males and females. (Read about Sexing Red-Eared Sliders for more on telling males from females.) When some male turtles try to woo females to mate, they approach them underwater and then one turtle will face the other and flutter or vibrate its front claws around the other turtle's head--right in its face! When female turtles catch sight of this and are amenable to the invitation, they drop to the aquatic floor.
At this point, they are ready to mate and fertilize. If a female is put off by all the fluttering, she may respond aggressively. Mating takes about 10 to15 minutes, but turtles can spend another 45 minutes beforehand just fluttering and wooing--think of it as foreplay!
Performing this courtship ritual does not necessarily mean mating will occur, though, and sometimes it thought to be more of a display of dominance or territorial type behavior.
Male turtles sometimes flutter their front claws in front of other males to express their higher social status. This is often an indicator that a physical battle is forthcoming during which the turtles might bite each other with their beaks (as they don't have teeth).
The turtles in this video are both males fluttering their claws and they are likely engaged in some sort of social dominance display since both are engaged in the behavior.
Sometimes young red-eared sliders will shake their claws around in an attempt at a wooing behavior, even though they're not ready to mate. Before maturity, the turtle can't breed successfully, but he can practice claw fluttering so he's ready when the time comes. Practice makes perfect!
Instead of fluttering, some turtles take a more gentle approach, using their claws to actually softly stroke the female's face rather than the maniacal shaking around their head. The male's claws, which are especially long (and noticeably longer than those of the females), are particularly suited to this special caress.