Designing a network is a topic within itself and there is no way we can cover all of it in this single post. Cisco has its own certification path path CCDE for these who want to know more.
Cisco design and implementation guide - old best practices
In a very simplistic view back in the old days a network used to be design similar to the picture below (although it is hard to say when the new era started ;)). Every big network had to have a core, distribution(sometimes called aggregation) and access layer. The network was engineered mainly to help with North to South traffic in the data center or in another words to help get the data out and into the data center.
New cloud friendly data center design best practice
With the advent and popularization of new networking devises that support layer 2 routing, commonly know as networks fabrics (more info about TRILL and fabric) the network design has shifted in data centers. The way we design the networks today is to maximize the East to West traffic instead of the North to South ( old design above). The purpose of the new network is to allow more efficiently exchange data between the servers within the rack or data center.
They say that a picture is worth more than a thousand words. To help us to visualize how a new data center network/cloud network is designed these demonstration pictures (taken from Cisco document: Cisco Massively Salable Data Center) will shed some more light on it. Please note that we no longer use the core, distribution or access keyword but instead: spine, leaf or superspine to describe the different network layers :).