I have been reading different stuffs on what a server means but i want a simple definition and the answers to the following questions.
I have always thought of a server as the CPU where other computers(monitors only) are connected to it and all the softwares and applications the user of these monitors use are installed on the processor server. (am i right?)
Must a server always be online?
Must other monitors connected to this processor server have their own processors or they all use this giant powerful processor?
This is what i always think of when i think of a server, please educate me.
Thank you.

There's no way you can talk about a server without mentioning a network. A network is a connection of computers communicating together. The role of a computer on a network determines whether it is a server or not. Thus, a server is any computer that has an important role it is performing to other computers on the network. For instance, a computer that assigns I.P addresses to other computers on a network is a DHCP Server, a computer that manage files on a network is a file server, a computer that has and manages the printer in which other computers on the network depend on is a print server, etc. Separate computers on the network can perform different server roles. It is however possible for a single computer to combine the roles. Whichever case depends on the choice of the network administrator. I hope i have satisfied you.

@buddy1960
Thank you. But i assume that network means the chain of connected computer. Does the network be connected to the internet?
What do you call this and is this following scenario possible. 4 display monitors in different rooms in a house connected to one processor but not connected to the internet. All the program this monitors display are installed on the one, central processors, so user in room 1 can open MS word, user in room 3 can open Autocad and on and on like that.
N.B: these monitors does not have their own processors. 4 monitors and 1 processors(CPU) not connected to the internet.
Is this possible and what is it called? Can one rightly call the processor a server? Thank you buddy1960.

@buddy1960
Thank you. But i assume that network means a chain of connected computer. Does the network implies it is connected to the internet?
What do you call this and is the scenario below possible. 4 display monitors in different rooms in a house connected to one processor but not connected to the internet. All the program these monitors display are installed on the one, central processors, so user in room 1 can open MS word, user in room 3 can open Autocad and on and on like that.
N.B: these monitors does not have their own processors. 4 monitors and 1 processors(CPU) not connected to the internet.
Is this possible and what is this type of structure  called? Can one rightly call the processor a server? Thank you buddy1960.

This is a server to me.

http://haymostech.com/TEST/SUNDAYFLEX.phtml

life_style32@yahoo.com

Network doesn't mean internet must to be involved. Network consist of at least two computers communicating together. Meanwhile, the fact that at least two computers are physically connected by either network cable or wireless doesn't mean they are automatically on thesame network. What really makes them to be on thesame network is the I.P. address configuration. If the connected computers have same network address, it means they are on a network. So it is possible for six computers to be physically connected and yet, more than a network exist in the connection. If the network involves internet, it is called an unlimited network. If no internet is involved, it is called a limited network or local area network.
For your other question, that kind of server is called a Terminal Server. When other computers on the network remotely connect to it, the server desktop will be displayed on the clients screen, thus clients will be able to run different applications installed on the terminal server.

^^^^^ Thanx Buddie.