The basics are that one device can talk to other device(s), via some shared hardware and software protocol. The protocol defines the language they speak so they can understand each other. The hardware defines the medium; whether they are talking using electricity though a wire or by using radio/light either through the air or via a light-guide (optical wire called fiber-optics).
Wired networks can be "shielded" from interference and noise easier, and so tend to allow higher speeds. But you have to physically wire facilities, which can be expensive, and then you're always tethered to the outlets and wires, which can be annoying. Plus, when wired technology changes dramatically, it may require rewiring; which is expensive. But with wired networks there is potentially more security, since in order to "break in" to your network, someone has to physically get at an access point, not just be in the proximity.
The way a network works, is that each node (device) on a network, can send and receive information. So each node is given an address; which works similar to your home address or phone number, it is just a unique identifier for that device. You obviously don't ever want two devices using the same address, because then everything gets confused over who gets what; so uniqueness of an address is important.
Addresses can be static or dynamic. Static addresses are non-changing, a network administrator manually assigns an address to each machine. Dynamic addresses can change, sometimes rarely, sometimes each time a user does something, depending on the network; but these addresses don't have to be manually assigned to each machine. Dynamic addresses are automatically figured out either by each machine (just asking all local machines what their address is and then using one that is available; AppleTalk and NetBUI do this), or more often there is just a machine, somewhere, which is the address manager. The manager gives-out a non-used address to each device that wants to use the network; so when my computer wants to use the network, it asks for (and is given) a unique address that it can use for that "session". Eventually, I turn that machine off, or enough time passes that the network takes that address back, and then that address is put back into the "available" pool. The Internet's (TCP/IP) dynamic address protocol (called DHCP) works like this.
Each device on a network is sharing the same medium (airwaves or current); so whatever a device sends, all the other devices can listen. So what the protocol does is setup rules for how they will all share that medium. Each device can only send a certain amount of information before they have to give other devices a chance to use the network as well, so there are size limits, and each chunk of information they send has to have both a target address and a return address so the network knows where the information should go, and that device knows where to return information back to. These chunks of data are called packets; and often when you send or receive a file, image, sound, etc., so it has to be broken up into many packets, and then reconstructed from parts on the other end.
Physically being on the network, and using the right protocol, lets a device (like the computer) talk to other devices on the network; but it isn't the whole story. While that gets the network working, we still have to do things like validation (account and password stuff), to know whether you have the rights to get the information you are asking for. And we not only have a device on each end, but sometimes they have many programs running, and we have to direct things to the right program; usually using a sub-address called a "port".
If you followed this somewhat dry explanation, then you have a fundamental understanding of what a network is doing, and theoretical understanding of how it is doing it.