VPS
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A Virtual Private Server (AKA VPS) is a level of webhosting service where you get root access to a virtual system and can install whatever you want.
For examples and more details, see webhosting#Virtual_Private_Server
IndieWeb Examples
Aaron Parecki manages several VPSs that host aaronparecki.com and many IndieWeb related services, as well as indieweb.org
Dr. Matt Lee runs his personal site plus Libre.fm from a series of VPSs from Bytemark.
Marty McGuire runs his personal site and various IndieWeb projects on a VPS.