First, you should note that mixins only exist in multiple-inheritance languages. You can't do a mixin in Java or C#.
Basically, a mixin is a stand-alone base type that provides limited functionality and polymorphic resonance for a child class. If you're thinking in C#, think of an interface that you don't have to actually implement because it's already implemented; you just inherit from it and benefit from its functionality.
Mixins are typically narrow in scope and not meant to be extended.
[edit -- as to why:]
I suppose I should address why, since you asked. The big benefit is that you don't have to do it yourself over and over again. In C#, the biggest place where a mixin could benefit might be from the Disposal pattern. Whenever you implement IDisposable, you almost always want to follow the same pattern, but you end up writing and re-writing the same basic code with minor variations. If there were an extendable Disposal mixin, you could save yourself a lot of extra typing.
[edit 2 -- to answer your other questions]
What separates a mixin from multiple inheritance? Is it just a matter of semantics?
Yes. The difference between a mixin and standard multiple inheritance is just a matter of semantics; a class that has multiple inheritance might utilize a mixin as part of that multiple inheritance.
The point of a mixin is to create a type that can be "mixed in" to any other type via inheritance without affecting the inheriting type while still offering some beneficial functionality for that type.
Again, think of an interface that is already implemented.
I personally don't use mixins since I develop primarily in a language that doesn't support them, so I'm having a really difficult time coming up with a decent example that will just supply that "ahah!" moment for you. But I'll try again. I'm going to use an example that's contrived -- most languages already provide the feature in some way or another -- but that will, hopefully, explain how mixins are supposed to be created and used. Here goes:
Suppose you have a type that you want to be able to serialize to and from XML. You want the type to provide a "ToXML" method that returns a string containing an XML fragment with the data values of the type, and a "FromXML" that allows the type to reconstruct its data values from an XML fragment in a string. Again, this is a contrived example, so perhaps you use a file stream, or an XML Writer class from your language's runtime library... whatever. The point is that you want to serialize your object to XML and get a new object back from XML.
The other important point in this example is that you want to do this in a generic way. You don't want to have to implement a "ToXML" and "FromXML" method for every type that you want to serialize, you want some generic means of ensuring that your type will do this and it just works. You want code reuse.
If your language supported it, you could create the XmlSerializable mixin to do your work for you. This type would implement the ToXML and the FromXML methods. It would, using some mechanism that's not important to the example, be capable of gathering all the necessary data from any type that it's mixed in with to build the XML fragment returned by ToXML and it would be equally capable of restoring that data when FromXML is called.
And.. that's it. To use it, you would have any type that needs to be serialized to XML inherit from XmlSerializable. Whenever you needed to serialize or deserialize that type, you would simply call ToXML or FromXML. In fact, since XmlSerializable is a fully-fledged type and polymorphic, you could conceivably build a document serializer that doesn't know anything about your original type, accepting only, say, an array of XmlSerializable types.
Now imagine using this scenario for other things, like creating a mixin that ensures that every class that mixes it in logs every method call, or a mixin that provides transactionality to the type that mixes it in. The list can go on and on.
If you just think of a mixin as a small base type designed to add a small amount of functionality to a type without otherwise affecting that type, then you're golden.
Hopefully. :)