You've got it right. In your example, "William" is the first name, and "Smith" is the last name. The first name is sometimes called the "given name" (because your parents choose this name and give it to you). The last name is also called the "surname" or the "family name." Even on formal documents, however, you will usually see the words "first name" and "last name."

Its a bit confusing really. William = forename or first name Smith = surname or last name Its very rare that you would hear someone refer to the word forename however - it will always be "first name". On the other hand, both "surname" and "last name" are used interchangeably.