Plants can move in a variety of different ways. Among them are moving from one location to another, rapid plant movement or nastic movements, and tropism.
In the first instance, many plants will change location during their germination stage. This is when the seed will leave the mother/father plant, usually by use of external means to spread the plant species around the area. Ways these seeds move are by having barbs that attach to the fur of passing animals (for instance grass burs), having "wings" or "propellers" that lift them with the wind (for instance dandelion fuzz). Another way seeds can be moved is when they are eaten by animals (such as apple or strawberry seeds) and then purged in that animal's waste at a different location. Creeping or English Ivy would be an example of an exception to the germination rule. Because it has aerial roots rather than ground roots, the ivy is able to spread over a large area without remaining anchored in one place. It slides along surfaces, growing where it can find purchase with its "feet."
Tropism is the process by which plants move in a particular direction due to outside forces. You see evidence of tropism when plants lean to a certain side or leaves turn to face the direction of the sun. This particular form of tropism is called phototropism.
Nastic movements are when plants react to outside stimulus, but it's not necessarily directional. Some plants react very quickly to outside stimulus. This movement is called Rapid Plant Movement. For instance, the Venus Fly Trap - a carnivorous plant - has hair sensors and when an insect comes in contact with it, they trigger the plant to close on the insect, capturing its prey. The mimosa, to protect itself from harm, will close its leaves and droop its branches when they are touched.