A package in Common Lisp is a set of interned symbols. In Common Lisp, systems are more in keeping with an ordinary understanding of packages...but combined with the idea of a build system just for fun.
ASDF is a way for managing systems (but it is worth keeping in mind that Common Lisp does not have any 'official' understanding of systems). ASDF is pretty much a de facto standard by consensus.
Quicklisp is a 'package manager' in the sense that it will go out and fetch a dependency from a repository. But what it fetches is a system: it is usually not a package in Common Lisp's technical sense.
From the Quicklisp FAQ:
How is Quicklisp related to ASDF?
Quicklisp has an archive of project files and metadata about project relationships. It can download a project and its dependencies. ASDF is used to actually compile and load the project and its dependencies.
ASDF is a little like make and Quicklisp is a little like a Linux package manager.
On the other hand, Common Lisp is very stable around ASDF and SLIME and QuickLisp. ASDF was started in 2002. Quicklisp in 2005. SLIME in 2003.