While I see that there's already an accepted answer, Common Lisp has some very nice documentation in the HyperSpec, and it's useful to know where to find the full description of what happens. In this case, it's 7.6.6.2 Standard Method Combination, which says (abbreviated):
The semantics of standard method combination is as follows:
If there are any around methods, the most specific around method is called. It supplies the value or values of the generic function.
Inside the body of an around method, call-next-method can be used to call the next method. When the next method returns, the around method
can execute more code, perhaps based on the returned value or values.
The generic function no-next-method is invoked if call-next-method is
used and there is no applicable method to call. The function
next-method-p may be used to determine whether a next method exists.
If an around method invokes call-next-method, the next most specific around method is called, if one is applicable. If there are no around
methods or if call-next-method is called by the least specific around
method, the other methods are called as follows:
All the before methods are called, in most-specific-first order. Their values are ignored. An error is signaled if
call-next-method is used in a before method.
The most specific primary method is called. Inside the body of a primary method, call-next-method may be used to call the next most
specific primary method. When that method returns, the previous
primary method can execute more code, perhaps based on the returned
value or values. The generic function no-next-method is invoked if
call-next-method is used and there are no more applicable primary
methods. The function next-method-p may be used to determine whether a
next method exists. If call-next-method is not used, only the most
specific primary method is called.
All the after methods are called in most-specific-last order. Their values are ignored. An error is signaled if call-next-method is
used in an after method.
If no around methods were invoked, the most specific primary method supplies the value or values returned by the generic function. The
value or values returned by the invocation of call-next-method in the
least specific around method are those returned by the most specific
primary method.
There's a particularly helpful illustration at the end of that page that describes the behavior and its motivation:
The before methods are run in most-specific-first order while the
after methods are run in least-specific-first order. The design
rationale for this difference can be illustrated with an example.
Suppose class C1 modifies the behavior of its superclass, C2, by
adding before methods and after methods. Whether the behavior of the
class C2 is defined directly by methods on C2 or is inherited from its
superclasses does not affect the relative order of invocation of
methods on instances of the class C1. Class C1's before method runs
before all of class C2's methods. Class C1's after method runs after
all of class C2's methods.
By contrast, all around methods run before any other methods run. Thus
a less specific around method runs before a more specific primary
method.
If only primary methods are used and if call-next-method is not used,
only the most specific method is invoked; that is, more specific
methods shadow more general ones.