Between 11:50 and 11:59 and until about 12:30 in the evening my computer begins to make a chirping sound that is repeated. At first I thought it was some type of alarm but now I am beginning to wonder. Any ideas?
Have you been playing Angry Birds? 
Have you turned on the monitor to see what’s running? Maybe a backup has been scheduled.
Every night? That sounds like a fan to me. Do you have a scheduled backup or something at that time that would make your machine go into high speed fan mode (maybe only applicable to laptops)?
I am usually using the computer. I have Avira, but that doesn’t kick on until 3:00 am. Definitely not the fan.
Is the sound coming from the speakers or the actual CPU? I have a computer with external speakers that, for some reason, will make an odd modulating, chirp kind of sound every time one particular cell phone in the house is getting a text message. I can actually predict the phone is about to ring with a new message in a few seconds if I hear the chirping sound start while I’m at that computer.
The computer has no speakers. It is coming from the computer itself. I am not sure, but it seems that the hard drive is running almost constantly at that point, that is the little lighted icon in front is steady, not flickering.
In that case it sounds like it is some scheduled activity taking place (backup, virus scan…maybe a malicious activity). It might be worth checking Windows task scheduler and see if there is anything in there set to run at that specific time.
If this is Windows, hit Ctl-Alt-Del, bring up the Task Manager, check the applications and processes. Then compare them to a time when there isn’t a problem. Sounds like something running and working the drive hard. The chirping could just be a lot of head movement. But something is causing it. Do you use Outlook? That could be doing an auto-archive. Lots of legit and illegit stuff could be going on.