A mainframe is a computer that is specialized in high throughput and transactional computation. That is to say, either the transaction succeeds and is committed, or it fails and is disregarded. There are classes of computation where this form of computing is required, for example, credit card transactions require that all-or-nothing computing transaction.
These computers are large, taking up several server cabinets all interwired, and they have thousands of processors and fiber optic wires – but not for computation, but just for moving data and organizing it for the CPU, to keep the actual central processor(s) saturated. Moving data from this side of the machine to that side of the machine takes time, and a whole lot of effort ($$$) is put in to keeping the data flowing as fast as it can. Any waiting for data to arrive is a failure.
These machines drive the most mission critical computing in the world, the most important transactions, the most important databases, etc. If you had to choose only one kind of computer were allowed to exist, civilization is built upon these machines.
A server is any computer that provides a service. That is, a request for a service comes in, and the computer responds. A web server is a kind of server. There are servers that issue IP addresses, and servers that map URLs to IP addresses… But while dedicated server computers are designed to handle high workloads and have high(er) reliability – not like your desktop (because $$$), they aren’t transactional in nature, they’re still just running programs, much like your desktop.
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