According to NLP adherents:
The NLP Eye movements indicate how a person is thinking - whether they are imagining a future or past event, internally re-hearing a sound or making up a sound, talking to themselves, or attending to their feelings.
For instance, when a person is accessing visual memories, they will characteristically glance upwards, while when they are accessing kinaesthetic memories, they will glance down and to their left. (Chart available at the link above.)
I've heard extensions to this that certain eye movements indicate lying (that is, some movement means "they're making it up" while others mean they are relying on memory), but to keep it simple, let's start with the basic claim. What does the science say?