Colocation
From Hackepedia
Colocation
Colocation means that your computer resides at an ISP and you pay a monthly fee for this with or without traffic charges. Colocations usually require you to have a 1U rackmount server so that you get 2 inches of rackspace for it, a desktop server is likely too big for the ISP to house.
Root Server
Some ISP's offer root-servers, meaning they provide the hardware to you from the start and you manage the Operating System when you have to. In Germany a root server costs around 30 to 40 dollars a month.
VServer
A cheaper alternative is running a vserver on ISP hardware. They use vmware or similar virtual engines to give about 8 people a server a host. A Vserver in germany costs around 7 dollars a month.