Bare-metal in this context means that you have physical hardware and you're responsible for making sure the system can boot and do the stuff you want, as opposed to what you'd have with a service like Amazon's EC2, where you're given a set of apps to configure and execute a virtual machine image. The distinction is made because the former scenario requires extra work for initial configuration (in terms of OS installation and physical networking and such) and you have the burden of setting up automation to handle scenarios where your OS installation is hosed, and much more.

Bare metal in this context typically means your servers are actual physical servers, not virtual machines.

As others have stated below, in this context you have bare metal and you have the cloud. You could also add in virtul machines, which run either on premise or in the cloud.

Thousands of physical servers our company manages that aren't rented, but owned by us. Does that help?

Yes. Thank you! If only industry marketing hadn't coined the terms, we might have: cloud -> provisioned server, bare-metal -> managed server or some other less context dependent terms.

Just for the curiosity, "managed server" in this area is already taken, and means that you get a sysadmin with the rented server (the server is managed for you, including installation and maintenance of the software you need on it). It is "higher level" than cloud servers, not "lower level" ;)