In Object Oriented Programming (OOP), inheritance describes a relationship between two types, or classes, of objects in which one is said to be a "subtype" or "child" of the other. The child inherits features of the parent, allowing for shared functionality. For example, one might create a variable class "Mammal" with features such as eating, reproducing, etc.; then define a subtype "Rat" that inherits those features without having to explicitly program them, while adding new features like collecting garbage.
If, however, you want to follow more than one totally orthogonal hierarchy simultaneously, such as allowing "Rat" to inherit from "Cartoon character" and "Disease vector" as well as "Mammal", without multiple inheritance you often end up with a very awkwardly mixed hierarchy, or rewriting functionality in more than one place (with the attendant maintenance problems).
Multiple inheritance has been a touchy issue for many, with opponents pointing to its increased complexity and ambiguity (for example, if two parents have independent implementations of a feature named X, which one do I inherit?)
Languages have different ways of dealing with these problems. Eiffel, for example, allows subtypes to adapt their inherited features by renaming them or setting selection rules for them ahead of time. Java allows objects to multiply inherit interfaces but only singly inherit implementations.