A broad term for a service where a third party "hosts" (physically stores) computers, content or services that are managed and/or maintained by the customer.
There are many types of hosting service employed today, including:
Dedicated hosting service, also called managed hosting service, where the hosting service provider owns and/or manages the machine, leasing full control to the client.
Virtual private server hosting, in which virtualization technology is employed in order to create multiple "virtual" servers that are leased out to customers
Colocation facilities, which provide the Internet connection, uninterruptible power and climate control, but let the client do his own system administration.
Cloud hosting, which can also be termed time-share or on-demand hosting, in which the customer pays for the system time and space used, and capacity can be quickly scaled up or down as computing requirements change.
Additionally, there are a number of popular, limited or application-specific hosting services, such as file hosting services, where customers store files on a service provider's machines, web hosting services, where a web page is hosted by a service provider, e-mail hosting service, where email services are hosted by a provider and DNS hosting services, where a service provider hosts DNS services and/or nameservers for a customer.