There are servers as in a computer. There are servers as in racks of computers in a room. Then there are servers as in huge data warehouses, such as Amazon and Microsoft and Rackspace and Google have. Those are usually called cloud-based computing. Most servers nowadays are virtual, in that many individual servers run on huge disks in racks in temperature controlled environments, and you can't run up to a box and go 'that one, Mumsnet's on there' because it could be anywhere on any SAN disk.
Anyone can run a server, they just need a server operating system and a pipe to and from the internet, and a DNS entry so the internet can find your computer.
Just as in desktop computers, you can buy different amounts of memory on your servers. You also pay for database licenses, although by the look of this site I think it's a LAMP tech stack which means open source which means free software, which is canny. There's also the bandwidth required to carry the traffic to and from the servers, this costs a fair bit of money too. Yesterday saw a massive spike in that traffic, and the pipe MN pays for will have been full up.
Companies usually have at least two servers and have a failover so if one server dies the other takes over, or so that the load is balanced equally between two servers. They have bought a certain amount of memory, processing power and bandwidth.
Most web hosting is set at a certain cost point with combinations of the above variables. After all, if you only ever get 300,000 hits a day, why would you pay for more on the off chance a thread with go viral. It's not worth paying out just in case you suddenly need twice your bandwidth/memory/processors for one day in a blue moon.
If you made it this far, go and get a gin down you, you must be bored rigid. :)