Identity is whatever makes an entity definable and recognizable, in terms of possessing a set of qualities or characteristics that distinguish it from entities of a different type (effectively, whatever makes something the same or different). Thus, according to Leibniz, if some object x is identical to some object y, then any property that x has, y will have as well, and vice versa (otherwise, by definition, they would not be identical).
Aristotle's Law of Identity (or the Axiom of Identity) states that to exist, an existent (i.e. an entity that exists) must have a particular identity. A thing cannot exist without existing as something, otherwise it would be nothing and it would not exist. Also, to have an identity means to have a single identity: an object cannot have two identities at the same time or in the same respect. The concept of identity is important because it makes explicit that reality has a definite nature, which makes it knowable and, since it exists in a particular way, it has no contradictions (when two ideas each make the other impossible).
Change is the alteration of identities, whether it be a stone falling to earth or a log burning to ash. For something to change (which is an effect), it needs to be acted on (caused) by a previous action. Causality is the law that states that each cause has a specific effect, and that this effect is dependent on the initial identities of the agents involved.
We are intuitively aware of change occurring over time (e.g. a tree loses a leaf). The Ancient Greeks took some extreme positions on the nature of change: Parmenides denied that change occurs at all, while Heraclitus thought change was ubiquitous.
Currently there are three main theories which deal with the problem of change:
Mereological Essentialism assumes that an object's parts are essential to it, and therefore that an object cannot persist through any change of its parts.
Perdurantism holds that objects are effectively 4-dimensional entities made up of a series of temporal parts like the frames of a movie (it treats the tree, then, as a series of tree-stages).
Endurantism, on the other hand, holds that a whole object - and the same object - exists at each moment of its history, (so that the same tree persists regardless of how many leaves it loses).