As Already mentioned, run a port scan and see what you have open, then you should run a vulnerability scanner for freeware see Nessus, if you want to buy something - and they are not cheap by any means - check out ISS Internet scanner, Eeye retina etc.. but I`d stick with Nessus if you can, although you need Linux to host to server portion. If you only have windows your freeware options are Sara or Saint (I think) or an early version of Cerberus, or Leviathan or Pluto. Run that and you`ll see the actual holes on your box. This will provide you with far more knowledge then a simple port scan, for example if you run a port scan on a box with Web server it will tell you that port 80 is open, but will not tell you if you are vulnerable to any of the Web server exploits out there.

Beyond that, get a firewall, Sygate, Tiny, ZoneAlarm etc... disable NetBios if you can, and apply any of the OS hardening techniques you find listed in any of the Hardening guides you can find on the net (see The NSA, SANS or MS for Windows, NSA or SANS for Unix) and that should give you a good level or protection.

Oh, and install some anti-virus software.

As for how hackers (or crackers) might access your system there are many many ways, and to go into them here would not be practical, check out some of the sites listed for details.

Good luck.