Okay, I'm back... I crashed on the couch.
A few questions:
1) do you personally OWN the computer (did you pay for it)
2) Is this a portable PC (laptop/notebok)?
3) Do you have Admin rights?
4) Does this computer need to be returned to the school at some point?
5) Are there known spy softwares on the computer (key loggers, website tracking softwares for example)?
6) Did you sign any type of release, or acceptable use form/policy?
7) Is there some acceptable use policy you automatically agree to by using the computer?
Now, Let me start my lecture
You will note, I do not tell you how to hide your tracks, not till you answer the questions I asked. I'm not the oldest one here (I think) but I can promise, too, I wasn't born yesterday. So don't lie to me either
If the computer is owned by the school (district), it is more than reasonable to believe that they have an acceptable use policy, and that they have EVERY RIGHT to enforce it.
I believe that the school fully expects you and others to download/use chat programs and other freewares (if it is a laptop). I don't doubt that they expect that you will install games also.
But I also expect that the end user of any computer should not violate any laws in its use (software piracy, copyright violations....). This is not unreasonable. And it is well within the school's right to demand renumeration for damage/legal fees incurred in the defense of any illegal activity. (charge you for the legal fees when RIAA sues because you downloaded music illegally).
I also expect that you did agree to an acceptable use policy whether in writing or not. Otherwise you would not have been assigned a PC.
There are steps that the school can take to ensure you don't do too much to the softwares. This can be managed by software like SteadyState and DeepFreeze, which after a reboot will return a PC to a predetermined software status (deleting any and ALL changes). Additionally, they can set other settings to allow easier monitoring, and expect that any changes you make are knowingly done to bypass their acceptable use, which in turn gives them the right to confiscate the computer from your possession/use.
This is all moot if you paid for the computer yourself, up to a point. If you connect your computer to their network, it is well within their right to enforce the same acceptable use of network resources. (no pornography, pirated music or softwares).
If everything you do is legal, ethical and moral, then you have nothing to fear. So why the guilty conscience? Again, I believe they expected some use of the computers. They may not like it, but it was expected.
Now I will not go into details on how they can track besides what has been mentioned already, but there are many ways. Your network sign-in is one way, the computer's information is another (even with spoofing it can still be determined).
If you are so worried, then turn yourself in. Ask for mercy, and don't do it again. You may learn a bit about your system in the process. Also, read those long boring acceptable use policies. I don't say re-read them, because I highly doubt you read them in the first place. That is human nature. That is why spywares get installed, that is why MS gets away with things like the Genuine advantage fiasco, people don't read what they are agreeing to.
Become Informed!! or as that one-eyed magic user would say... constant vigilance.