In its simple form, 'the cloud' refers to servers that are accessed over the internet, and the software and databases that run on those servers. These servers are located in big data centers all over the world.
In other words, the cloud replaces the need for having your own servers and running business critical applications on your own infrastructure. In a cloud world, you effectively outsource this to large technology vendors like Microsoft and in return, you get a lot of business benefits.
It's like renting a house: The landlord own the house (the servers) and takes care of all the maintenance, but what's inside the house (the applications and data) is up to you as a tenant.