What does it mean that Javascript is a prototype based language?

Prototypal inheritance is a form of object-oriented code reuse. Javascript is one of the only [mainstream] object-oriented languages to use prototypal inheritance. Almost all other object-oriented languages are classical.

In classical inheritance, the programmer writes a class, which defines an object. Multiple objects can be instantiated from the same class, so you have code in one place which describes several objects in your program. Classes can then be organized into a hierarchy, furthering code reuse. More general code is stored in a higher-level class, from which lower level classes inherit. This means that an object is sharing code with other objects of the same class, as well as with its parent classes.

In the prototypal inheritance form, objects inherit directly from other objects. All of the business about classes goes away. If you want an object, you just write an object. But code reuse is still a valuable thing, so objects are allowed to be linked together in a hierarchy. In javascript, every object has a secret link to the object which created it, forming a chain. When an object is asked for a property that it does not have, its parent object will be asked... continually up the chain until the property is found or until the root object is reached.

Each function in JavaScript (which are objects themselves) actually has a member called "prototype", which is responsible for providing values when an object is asked for them. Having this member allows the constructor mechanism (by which objects are constructed from functions) to work. Adding a property to the prototype of a function object will make it available to the constructed object, as well as to all of the objects which inherit from it.

Advantages

There may not be a hard and fast rule as to why prototypal inheritance is an advantageous form of code-reuse. Code reuse itself is advantageous, and prototypal inheritance is a sensible way of going about it. You might argue that prototypal inheritance is a fairly simple model of code reuse, and that code can be heavily reused in direct ways. But classical languages are certainly able to accomplish this as well.

Sidenote: @Andrew Hedges makes a good point, that there are actually many prototypal languages. It's worth noting that these others exist, but also worth noting that none of them are anything close to mainstream. NewtonScript seemed to have some traction for a while, but died with its platform. It's also possible to extend some modern languages in ways which add prototypal capabilities.

A prototype-based language, does not make the distinction of classes vs objects: it simply has objects. A prototype-based language has the notion of a prototypical object, an object used as a template from which to get the initial properties for a new object. Any object can specify its own properties, either when you create it or at run time. In addition, any object can be associated as the prototype for another object, allowing the second object to share the first object's properties.

Prototype-based programming is a style of object-oriented programming where classes are not present, and behavior reuse (or inheritance in class-based languages) is performed by cloning existing objects that serve as prototypes.