ok...
servers first...
fundamentally there is nothing "special" about a server.
yes, servers can have more than one processor, more than two, the most processor sockets that I've seen recently on some of the servers we use at a customer is 4, that's 4 sockets with quad core chips in each making 16 cores.
RAM, servers tend to have more slots available for RAM, instead of 2, think along the lines of as many as 16 slots.
Network, lots of servers have two or more network ports on board, (of the 1000MB variety).
power, -lots of servers are going to have two power supplies so that they are redundant (one can fail and the server keeps going).
Disks, your typical server is going to have very very fast disks, and probably lots of them, you're also likely to see some decent hardware RAID in a server such that your system drive is actually stretched across multiple drives.
Case - most servers that you intend on hosting in a data centre are going to come in rack mount cases, simply to make it a lot easier to mount the server in a rack, on it's own.
out of band management, most servers are going to have what's called out of band management, for HP systems this is usually called ilo, for IBM systems it's RSA or IMM, these allow you to control the server remotely, you can log into a web interface to shut down the server, look at hardware events, monitor the temperature, possibly even use a screen that is a console connection, (i.e you see what's on the screen at the server), you might even be able to mount iso images as virtual devices to rebuild servers remotely, (a good thing when your server is in London and you're in Oxford, as some of our servers we look after are, absolutely essential when you're in oxfordshire UK and your servers are in Virginia in the US, as some of our other servers are, then it changes from an inconvenient bus ride to rebuild a server to a very inconvenient flight, and at the moment with UK air space closed and an ash cloud moving closer to America, it's practically an impossibility, a few days sailing? a drive round the north circle to re-install windows? -no thankyou!!
All that said, there is nothing special about a server really, it's still just a box, with processor/Ram/disks and any machine can be a server.
The next part is somewhat more complicated...
why is hosting a server expensive...?
that's because you're renting space in a rack in a data centre, or you're renting entire rooms of racks, that may be physically separate rooms (as we have on one data centre) or metal cages with racks in them that keep your equipment secure, (as we have in some other data centres).
Data centres tend to be built in quite built up areas they will usually have many many connection links from just about every telco that you can think of, (including ISP only people like level 3 and others). so you're looking at buying premium space, think prime real estate type buildings (even they may be purpose built or just old warehouses), connected to multiple parts of the electricity grid.
also data centres, even though they are connected to multiple grid points, also have to have instant on backup generators. that are able to power the whole building. your rent is going to be paying for parts of all of this.
then you'll also be paying for a high speed line in/out of your server, even if that's a shared line with other customers it can be expensive...
the thing is this,
if you want to host your own personal website, that won't be receiving much traffic, then take the hit on your home electricity bill and host your server from home on your home broadband connection.
if you're planning on hosting a game server or high traffic websites then you're going to need to get a much faster connection, the kind that you're only really going to find in data centres that connect straight into internet back bones.
and when you start hosting in a data centre your going to pay for the space that you use, and the power that you use.
so you may save a thousand dollars by reusing an old tower case, and an old board, old processor, spare RAM etc, but you're going to find that your case takes up ten or 20 rack spaces, so you're going to pay for half a rack (which will be expensive), your components won't be efficient, so you'll pay the same for power, even though your machine won't have as much processing power.
(i.e you're probably going to loose your saving in the first year!).
you'll also need to be able to visit the data centre easily if you need to put in a CD or rebuild the machine or something since you won't have the out of band management.
So the long and the short of it is this:
you can host at home, and this will only cost you the price of your own electricity bill.
but your sites that you host will only be as reliable as your own electricity or internet connection. and your site sill only host as fast as the upload speed of your internet connection... and this will be contended, (think about it, say that you're running your site, someone connects they want to download a page from your home hosted server, but they can only use as much bandwidth as is available if you're also using the internet you're sharing that bandwidth with your site visitors. if you download something using a torrent program, you're likely to saturate your upload bandwidth and now your own site off line!
-i.e you might find it difficult to sell such low quality server space to others to host their stuff on.
You can build your own server, but if you do that you're going to loose some advanced features of a real server, (many processors, more ram redundant power/network remote access).
you can host in a data centre, then you can almost guarantee complete up time for power and internet connection, but your going to pay a premium for this. (one that you may not be able to recover with just one server).
