Glad to hear you're not intimidated by a computer. Not so glad to hear it's not giving you the respect you deserve. When a computer fails, it should fail and when it works it should work. It's the on and off bit that's frustrating.
I do not know your E-machine so I can't make any model-specific comments, but at 5 years any computer is getting on a bit, and may be developing eccentricities... Also any new item is itself suspect, that's why a warranty is important. That aside..
You can check your system's BIOS. When Windows can't see the disk - does it show in your bios? If it does - you have a Windows problem (and advising on that's beyond me). If it doesn't show - it's a hardware problem, and Windows doens't show it because the hardware can't find it.
An act of desperation you can try: if your drive is set to AUTO in hte BIOS, you might be able to change it to the drive's specific configuration (which is usually detected autiomatically). Or if it is specifically configured, you might change it to AUTO. It may make a difference.
Then I'd check the IDE and power cables are seated properly, and check that when the disk is not recognised, that it is spinning (possible power connector problem). It is also possible your IDE cable itself is acting up. If you have a spare available it may be worth a try.
You can usually hear a drive spinning, but if the computer is loud enough to mask it, you can use a screwdriver as a stethoscope - put the end of the handle against your ear (yes wash it first) and the tip against the drive case. Avoid the circuit board! If you find it is not spinning you may have a weak power connector. You can try a different connector - perhaps the CD drive's connector.
Next - check the original drive's jumper settings. Most drives have master/slave/CableSelect, but some vary. Some older Western Digital drives, for example, have 3 settings - single drive, master with slave, and Slave - as well as CS. If your original drive has a 'single master' setting it needs to be changed to recognise slave properly. You can get the jumper settings from the disk maker's web site.
Most likely, however, the original disk is set to "Cable Select." Cable select is a way of assigning master/slave depending on which connector goes to the drive - end is master, middle is slave. I think. Also, the middle connector is not exactly in the middle - the longer side always goes to the motherboard. (Maybe this isn't always true with custom cables in a name-brand computer.) A Cable Select IDE cable has one of the wires cut. You can't usually see it, but this is very convenient for manufacturers - the disk/CD/DVD etc makers set the jumpers to CS and the system builder just takes the required drive and plugs it in. Nothing to configure, nothing to get wrong, and a few seconds saved. And time is money. (insert maniacal laugh here)
SO as an upgrader, you have to check the existing drive - is it set to master (or single drive) or CS? - and then you have to decide how you want to configure... You can set your drive to CS and connect to middle - and it is recognised as slave. If it is physically inconvenient to connect your new drive to the middle, you may prefer to re-jumper your exisiting drive to master, and jumper your new drive as slave, and then it doesn't matter which connector you plug into which drive. However - you should not have one drive set to CS and the other to slave.
OTHER THINGS YOU CAN DO.
As a diagnostic tool - try disconnecting your CD/DVD and hook that IDE cable to your new drive. If it sees that reliably you know it's not the new drive. If it still plays up - you likely have a dodgy drive.
Instead of having main drive/backup drive on the same cable, put your backup drive on the same cable as the CD/DVD drive. This will likely give you slightly better performance when making backups. Before you do this, make sure the CD's IDE cable is an 80-wire cable or the hard drive will be very slow. Many makers put an 80-wire cable on the hard drive and a 40-wire cable on CD, since the CD is slower and doesn't need the speed an 80 wire cable supports. It saves a few cents. It is easy to see if the 2 cables are the same or different - just look at them. If your CD cable is a 40-wire cable, you can change it to an 80. That will let the disk run fast, and the CD won't care.
Maybe external USB is the best option for a backup drive anyway. Any drive in the case could get fried when the main drive gets fried... power surge, lightning, power supply failure, virus attack... If you unplug it when you're not actually making backups or doing a recovery, you elimiate (minimize) these risks. Of course there is always the risk of dropping an external drive. At 5 years old your computer probably has original USB, limited to a little over 1 MB/second. This makes an external drive painfully slow. You can find a USB2 PCI adapter card that gives you about 48 MB/second transfer online for around $10 (buy, newegg, many places). Offline - expect to pay $30.