TL;DR version:

• It does still exist.
• It has become a lot less prevalent, but only partly due to post-911 security changes.
• The savings are far smaller than they used to be.
• They use your checked luggage, not your hand luggage.

Sources:

The term used is air courier, and I found several resources on the topic after more searching:

• WikiVoyage actually has an article on air couriering.
• about.com has a good article too.
• There's a writeup here on a website called seminar.org.
• A company called "Air Courier International" has information specifically detailed for the UK and USA.

The term used is apparently air courier. I've found several resources on the topic after more searching.

But I have not found the names of any courier companies. They seem rather elusive!

TL;DR version:

• It does still exist.
• It has become a lot less prevalent, but only partly due to post-911 security changes.
• The savings are far smaller than they used to be.
• They use your checked luggage, not your hand luggage.

Sources:

• WikiVoyage actually has an article on air couriering.
• about.com has a good article too.
• There's a writeup here on a website called seminar.org.
• A company called "Air Courier International" has information specifically detailed for the UK and USA.