OK, so you have a computer. . . Does it make sound? Does it have speakers, either built in, or sitting on the desk beside it? Have you ever played music on it? Does it have a place to plug in headphones/earbuds? Almost all computers nowadays have these things, but you need to be sure. Do you have headphones or earbuds or such that you can plug into the computer and listen to music on? If not, you need to go get some. $10 at Best Buy or Radio Shack will get you a set of earbuds, a couple bucks more for cheap headphones. I think cheap earbuds sound better than cheap headphones, but I can't back that up. You will need these for hearing the music while you are singing.
I will assume from here on that you can get sound from your computer, ie. if you go to youtube and play a random video you get sound with it. I will also assume that you can put your earbuds in and hear that sound only in your ears, not out in the room.
Ok, so we can get sound out of your computer, and we can put that sound into your ears while keeping the room quiet. Next order of business is to make sure you have a way to get the sound of your voice into the computer, which is to say... Do you have a microphone? If yes, does it have the right size connector on it so that you can plug it into the computer? On the computer, you're looking for an input that says something like Mic In, or it may have little picture of a microphone. If your mic has the right size connector to fit into this jack, it is probably the right kind of mic to use without anything else between the computer and the microphone, so go ahead and plug it in. We'll test it in a minute. If you don't have a microphone, does your computer have one built into it? Most laptops do and some desktops. Neither the microphone you can plug in directly nor the one built into your computer are suitable for high-end recording, but if you're just looking to make a simple recording as a place to start either one will work.
Now to test whether your mic is working... I'm not sure if windows xp is the same for everybody, but on my machine you would: Click "Start" in the lower left corner of your screen. Select "Programs" from the menu. Click "Accessories" in the popout menu that appears. Click "Entertainment" from the next menu, and finally click "Sound Recorder" from the next menu. A little box will pop up with record and playback controls. Click the "Red Dot" button to start recording and start talking (into the microphone if you have one. Just into the air if using a built in mic). After saying something, click the "Black Square" button to stop the recording. Now plug in your earbuds and click the "Single Black Triangle" button to play back the recording. Do you hear yourself? If yes, your in business. If not, on the menu bar at the top of the little recording box, click "Edit". In the menu that drops down, click on "Audio Properties". The audio properties box will pop up. In the middle of this screen there's a box where you can select your audio input source. If it is blank, open the list by clicking the little black triangle just to the right of the box. If the list is empty, it's the end of the line for my instructions. If not, select an option from the list. If there's more than one option in the list you may have to try them one by one til you find one that works. If the box wasn't empty to begin with, or you were able to select an option (or a new option) from the list, now click on the "Volume" button just below the selection box. Another window will pop-up listing the actual inputs for the selected audio source. Typical examples would be line-in, internal mic and external mic. If you are using a plug in mic, and there's also a built in mic, this is where you would pick the one you want to use. Each input will have a slider for adjusting the sensitivity of the input (for controlling your recording level, just like on your old reel to reel machine). Below the sliders there will be little square check boxes for selecting which input to use. You can only check one at a time. Check the appropriate one and repeat the test above.
Once you have succesfully recorded a sound and played it back, you'll know your hardware is capable of doing what you want to do. Now you just need to have some software to work with. The purpose of the software will be to play your background track (like the karaoke machine would) which you will listen to in your earbuds, and at the same time record your voice as you sing along. This is the step where you will need to adjust the sensitivity of the mic for the loudness of your voice. After you've recorded your singing, the software will let you play back the karaoke track and your recorded voice at the same time and adjust the relative volume of each. This kind of software is called a Digital Audio Workstation or DAW. Most of the ones folks talk about in this forum will do lots of other things, but at the heart of them all is the ability simultaneously to play back the background track (or previously recorded tracks) and record the musician(s) or singer(s) as they play or sing along in time with the music they are listening to. As for knowing when to sing and when to wait, it is just a matter of knowing the song well enough to follow along with the sound of the music, or counting out the beats during the parts where you aren't singing as several others have commented above.
As for which software to use. To my knowledge, windows XP doesn't come with anything that will work for what you want to do. There are two shareware/opensource/freeware packages that I know of. The first is called Audacity, which you can download here:
Audacity: Download
The other is called Reaper (the one I use), which you can download here:
REAPER | Download
There are also many you can purchase, and there may be even simpler free versions you could download for your purposes. Downloading, installing and setting these up is beyond what I can type here, but if you can get through the test listed above, any of these software packages will do what you want to do (albeit with limited fidelity depending on the microphone you are using). The only remaining thing you would need would be the actual soundfile you want to sing along with. Could be for example a karaoke machine disc.
Good luck,
J