Restored Computer, Missing Drivers, Cannot Connect to Internet

You almost have it.

The TEW-623PI is a TRENDnet PCI wireless adapter.  The only problem is that there are two different versions, which require different drivers.  Look again and see if you can see whether you have Version 2.0R (info and driver download HERE) or Version 3.0R (info and driver download HERE). 

The installation for these files is quite a bit easier:  there's a Quick Installation Guide (scroll down the page and click the link).

Actually, the Quick Installation Guide wants you to install the driver BEFORE the device is physically installed in the computer.  So, turn the power off, touch the case with your hand to discharge any static electricity, and carefully remove the card.  There probably will be one screw holding the card in (if the screw is present, it will be in the slot at the top of the bracket -- see arrow):

Then gently but firmly pull the card up. It probably will be easier to remove if you unscrew the 3 antennas.  You may have to pull one end of the card up first; just don't rock the card side-to-side.

Once the card is out of the computer, you should be able to more easily see if you have version 2 or version 3.  And, while the card is out and you have more room, do yourself and your computer a favor:  buy a can of compressed air and blow out all the dust.

The download from Trendnet installs a configuration utility for the adapter.  To learn how to use this, download the User Guide, link at the bottom of the support page.

Ok so the thing that I'm pretty sure is the wireless adapter is triangular, but I can't read anything very well because it is very dusty and almost directly beneath another component.  The only thing I got off of it was tew623.  Sorry I can't be more specific about that.  When I tried to update the Network Controller driver it said that I didn't have the necessary files or something.

Doing things blindly is probably not the best option, but short of having someone with some familiarity with computer components look inside your computer (or asking your friend for the make & model of the wireless adapter), the following is about all I can think of to suggest.  The problem is that although other manufacturers of wireless adapters use the same Ralink 2860T chip that's in the WMP600N (for example, this Rosewill RNX-N300X b/g/n wireless PCI adapter), it's not at all clear that they will all use the same driver provided by Ralink or if they will modify the driver to deal with additional hardware they have included in their products.

First, just so things are clear, what you would be looking for is something like this:

Linksys WMP600N

or, because you have 3 antennas, this:

D-Link DWA-552

Do the following with the understanding that it may work, work partially, or not at all.  You really should make an effort to determine the make and model of your device.

Right click on the "Network Controller" in the other devices category.  If it has an option to "Update driver software" (or something similar), select that and follow the directions at the Linksys article I mentioned earlier.

In the other devices category I have Multimedia Audio Controller, Network Controller, and Video Controller (VGA Compatible).  I looked inside the case and frankly It's kind of a mess in there. I looked around on the inside where the antenna come out the back and I didn't see anything that resembled a model number or even a card.

Sorry.  This was my error.  Because it has no drivers, the device with the antennas (your wireless network adapter) is in the "Other devices" category rather than the Network adapter category.  Your conversation with Xircal started along the road to identifying it, but you got sidetracked with the discussion about the motherboard.

The 3 antennas definitely suggest that you have a wireless adapter.  However, the Linksys WMP600N only has two antennas ... so that's not what you have.  Nevertheless, it's possible that the drivers for that device will work with your adapter because they apparently have the same chip.  It would be better, however, to get the driver that is specifically intended for the hardware that you have.

The "1394 Net Adapter" is a FireWire adapter.  This is hardly used any more.  It was Apple's attempt to provide an alternative to USB.  LINK

The "NVIDIA nForce 10/100/1000 Mbps Ethernet" is the Ethernet adapter (built into your motherboard).

Neither of the above is a wireless adapter.

Please list all of the items in the "Other devices" section of Device Manager. 

I don't know if Device Manager will provide enough information about the "other devices" to identify the make and model of the wireless adapter.  When you said that you couldn't confirm that you had the Linksys adapter, did you open the computer case and look at the card?  If not, that's the best way to identify it -- there should be a label somewhere on the card. 

If all else fails, perhaps you could call your friend and ask what type of wireless adapter he installed.