Algae growing on a turtle's shell are usually not harmful. Most wild turtles have some algae growing on their carapace, some of them have quite a lot, and it makes them hard to see in a pond. Which from a wild turtle's perspective, is a rather good thing.
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, but 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 in the tank 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. Keeping the water moving will also make it harder for the algae 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. Do not use!
To remove algae from the turtle, brush it with a SOFT brush. A 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).
If algae grow in your rocks and tank decorations, unless they disturb your sense of beauty, you can let them grow. An exception is the long, hairy kind which also makes the water slimy. Those, you should remove. Algae grow in healthy water with enough light. They are a sign, that you are doing something right. Do not use chemicals to kill algae!!! If you don't like the algae, brush them off every time you change the water, change the water more often, use a stronger filter, and add a little salt to the water (see further up). In the wild, it is normal for turtles to grow algae on their shells. It helps them camouflage! In captivity, the algae should be removed every once in a while, since they can encourage growth of fungus in a confined environment. To remove the algae, hold your turtle under warm tap water and gently brush it with a soft vegetable brush.