Topology
Our network is using what is called Star topology with the router as the centre (server) of the Star.
All the other computers (clients usually )are making up points of the star. This is most common for small to medium sized networks (up to 50 machines or so) .
Its good because each machine gets is own dedicated network connection and if there are any cabling problems, for example a rat chews through a cable, it only affects the one machine. Everyone else caries on happily except maybe for the rat.

You start to get problems with a star network as the number of points on the star (clients) increases. This is because there is a maximum length a network cable can be and there is also a maximum number of machines (clients) each sever can deal with.
On larger networks a Bus topology is often used.
The bus on the diagram on the right is the horizontal line. This is often an upgraded connection, these days typically fibre optic cable.
The other thing to do is position a few servers along the bus to cut down the work each one has to do and so speed up the network.
As networks get bigger still they often become a mix of star and bus with "stars" on each node off the "bus" or backbone as it is sometimes called.

Clients and Servers
When talking about clients and servers it is important to understand what each is and that some computers can depending on what job they are dong be both at the same time.
Servers
The router on my network only ever works as a server, it provides services. It can give you an internet connection it can give you a name for your computer on the network. It can give you access to a wireless connection. It can make sure you see the webpage you asked for and it can give you protection via its firewall.
So far so good.
Clients
Most desktop computers most of the time are clients. Running a web brower (internet explorer) for example is a client activity.
You computer uses some of the routers services, the connection to the internet and its ability to make sure you get what you asked for, to display or serve up the webpages you ask for.
When I turn on my wirelss laptop and it automatically connects to the network it uses the routers' wirless services to gain access. It is a wireless client.
The dsm 320 is an exclusively client device as it can only playback files (mp3 or video) and as it has no storage of its own. Without the services provided by the music server in the hall it would look quite nice, but it would be an expensive ornament.
Again this seems OK.
When Clients are Servers (when it gets a bit confusing)
On my network the printers are shared by rachael's iMac this means the iMac controls the fact that other computers on the network can see the printers and more importantly can print to them.
The iMac maintains queues of documents to each printer and if the imac is turned off you can't see the printers from other machines on the network. It is important to understand the iMac when it offers these services is working as a server. A print server in this case.
When my pc in the hall is making available my music library to the dsm 320 in the living room it is a server. These days quite often most desktops in smaller networks are working as clients and servers at the same time depending on what is happening.
The client/server model is vital to an understanding of computer networks and the internet in particular .

Remember
Servers give you stuff
Clients just take things