Asperger's is a kind of autism, but not all autism is Asperger's.

Kind of the way an orange is a kind of fruit, but not all fruit are oranges.

Asperger's is basically what they call autism without a speech delay or developmental delay. It's categorized separately because it was first described by a guy named Hans Asperger; but his work was lost for a good while. Separately, a fellow named Kanner also described autism for the first time (using, ironically, cases of which some could be called Asperger's today); and his was the definition people used until Asperger's work was rediscovered by Lorna Wing, who popularized Asperger's as a concept in the 1980s. By then, "autism" proper had speech delay as an intrinsic feature, so they were named separate diagnoses.

Forward to the 1990s and a new DSM edition. Asperger's is now entered into the DSM as a "pervasive developmental disorder", which is the new name for the entire autism spectrum, and autism in general becomes much more popular as a diagnosis as it is found in people previously thought to be (or only to be) mentally retarded. Asperger's also becomes more well-known, and it expands the spectrum to people whose major problems are with repetitive movement, constricted interests, and social interaction, rather than speech. With this very broad spectrum, autism is now known to be very common.

Enter "PDD-NOS", the diagnosis for people who are on the autism spectrum but don't fit either Asperger's or regular autism. It is currently the single largest category for autism, and includes people with very diverse features. "NOS" is the name for "not otherwise specified"--when they can't find a diagnostic category--and it's supposed to be a small minority, pointing to the inadequacy of the current criteria. As if that weren't bad enough, lately it's become obvious that Asperger's and the rest of autism aren't different enough to put them in separate categories (Asperger's speech is often highly unusual, and regular autism is often indistinguishable from Asperger's in adulthood), so they're thinking about just merging the entire autism spectrum and calling it an "autistic spectrum disorder" in general.

So... My recommendation is just to consider Asperger's as a vague patch somewhere on the autism spectrum. It's not a really well-defined group, and is probably not a distinct condition; but it has definitely served its purpose to teach us that autistic people can be highly verbal and do not have to have developmental delay.