If Office 365 is like an Exchange Server in the Cloud, what do you do if a particular application server needs upgrading, or you need a server for a new application but the users that will access it will be located in different locations across Europe.
A cloud server can come in one of two forms, a server configured and managed on the Microsoft Azure or AWS platforms, or one that is hosted in a UK datacentre.
The two models are essentially the same in how they are provided, they both involve massive datacentres with virtual servers sitting inside server farms attached to pools of large Storage Area Network devices. These datacentres are designed so that disks can fail, connections can fail and servers can fail but your server just keeps on going.
The key difference is that; with the Microsoft/Amazon model, you have no real control of where that server and more importantly your data resides, whereas with the hosted server, you know that it is in a certain datacentre in a certain location.
In either case the model is a subscription one, so you can choose to run just the server you need today and upgrade that server and the connection to it as you expand.