Algae or Slimy Film on Shell

Algae growing on a turtle's shell are usually not harmful. Most wild
turtles have some algae growing on their carapace. 
However, the algae might hide some potential shell
problems, especially in animals whose shell is already stressed and
pitted from previous disease.
Occasionally, the algae will grow under partially shed scutes, then water
accumulates there, and a local shell problem might develop. This is rare. 

If your turtles are growing algae on their shells, inspect them
thoroughly once a week to make sure you are not missing any potential
problems. (You should be doing this anyway!)

Algae also have beneficial effects: they help keep the water clean
and healthy! I usually let them grow a bit in the tank for a less
sterile appearance.

In order to grow, algae need some sunlight and nutrients. Deprive them
of both, and your problem will lessen or go away. Also, keeping the water
moving will also make it harder for the alge to take hold. 
To prevent algae growing on a turtle, change the water frequently,
and brush off any algae growing in the tank.
Don't expose the tank to too much sun. Adding a bit of
salt to the water every other week, to change the composition of
the water, can also prevent algae growth.
Aquarium stores sell chemicals to kill algae. These are not recommended.

To remove algae from the turtle, brush it with a SOFT brush. A discarded
toothbrush works really well. Usually, not all algae will come off the
first time.

There are many different kinds of algae. Some grow in clean, healthy water, 
and some grow in muck. As a general guideline: Algae that look dark green and
that grow in carpets or patches are fine. Long, stringy, slimy algae are not.
If you get the latter kind, change the water more often or get a bigger filter.

If the shell of your turtle feels slippery, but no algae are growing on it,
it is probably a film of bacteria. This is not good for the turtle. As a 
first measure, you want to brush the turtle's shell off with a soft brush; for
example an old toothbrush. Keep doing this weekly, until the problem goes
away. If you don't, these bacteria will get into any imperfections and weaknesses
in the shell, and they will cause shell infections. 
This kind of film usually occurs, if there are too many bacteria in the water,
and/or the turtle does not get to bask enough.
Change the water more often, get a bigger filter, make sure the turtle has a 
warm basking spot to dry out (drying out will kill the bacteria).