Wow! There's a question.

Well, it actually quite a mouthful.

The first step usually intells someone getting your IP address (ie: 216.234.23.9) and trying to connect to your computer to view your ports.

How do they get your IP?
Well there's programs like ICQ or mIRC that gives that info away in chat sessions.
There are also people who IP Scan (using a program that searching a range of IPs for a running machine - useful on networks with set IP)
As well as other means (google: keyword "find ip", "locate ip", "find target")

What are ports?
Ports are on your computer for "services" that are run.
The ports "hackers" use are virtual, as in they don't physically exist.
Examples of "services" ports run are like port 80 for http connection (internet) or 25 for smtp (e-mail)
What hackers do is scan those ports and find which are open and try to compremise them with known vulnerabilities. (google: "windows9x vulnerabilities", "linux vulnerabilities")

What's a vulnerability?
It's a programming error or a flaw in the code.
What happens is that a hacker does something with the service which is not intended to do and when it runs the "problem" the vulnerability will either crash your computer or open it up for access.

There's also Trojans!
Trojans are computer programs that makes a connection from your computer to someone elses. Usually, these are transfered by e-mail or chat apps like ICQ or Messengers. Someone runs it (not knowing what it is) and then it opens up your computer ports (usually above 2000 me think) to connections by other persons. Popular trojans are NetBus, Sub7 and BackOrifice.

What about NetBios.
While this isn't really a hack, it can give out information the attacker needs.
When you enable file&printer sharing it allows others to connect to your computer and view/copy your files to your select shared files (*depending on security mesures in place)
If you have a password file or something saved in that directory, the snooper can see it and use your accounts.

There are other things too, programs and such, but others here can give you move info about it. Also check sites like hackers.com and hackersplayground.org (while it might be a little much/confusing on information, it is sites like this that will give some light)

Also check this sites tutorials and forums for such topics. If there's something that confuses you about the topic, Prvate Message the author for more explinations

Hope this helps!