Symbiosis is simply defined as a very close relationship between two different species of organisms. There can be many different types of symbiosis.
In some types of symbiosis, the two species both benefit from the relationship between them. This is called mutualism. An example of this is the relationship between some species of wrasses and other fish. The wrasses "clean" the other fish, eating parasites and other things that irritate the other fish.
Another kind of symbiosis is commensalism. In this case, one species benefits and the other is unaffected. An example of this would be a remora attaching to a shark and getting food from what the shark eats.
The marietta.edu link below provides many more examples of various types of symbiosis.
Symbiosis is a close association of two organisms over a long period of time which may be beneficial for both the organisms, help one of them but not affect the other or help one of them and harm the other.
An example of a mutually beneficial symbiosis that has been going on for millions of years is that between insects for example bees and plants. Plants need to have their pollen propagated, so they produce nectar in their flowers. The insects feed off the nectar and propagate the pollen which sticks to their body.
Another example is between that of plants like acacia and ants. The acacia releases drops of fluid that contains sugar for the ants to eat, and in return the ants protect the trees against other predators by attacking them whenever the predators try to target the trees.
When two organisms live together, as if they are parts of the same plant, and are of mutual help to each other, they are called symbionts, and the relationship between the two is expressed as symbiosis. Lichens are typical examples. These are association of algae and fungi, and commonly occur as thin round greenish patches on tree-trunks and old walls. The alga in a lichen being green prepares food and shares it with the fungus, while the latter absorbs water and mineral salts from the surrounding medium, and also affords protection to the alga. Another example of symbiosis is Rhizobium radicicola bacterium. These bacteria have the power of fixing the free nitrogen of the soil air in the nodules. The leguminous plants supply the bacteria with carbohydrates and the bacteria supply the former with nitrogenous food.