Hello Melone06 and welcome to the forum.
Here's a rough description of what must be done :
Pull out plug from computer (no, shutting the computer down is not quite enough)
Open case of computer (this varies from system to system so I can't tell you how exactly).
Disconnect (pull out) two cables from the back of the DVD writer, one with a short connector with 4 wires (red, yellow, black usually) coming out of it (power cable) and the other with a longish connector connected to a flat ribbon cable (data cable).
There may also be a single wire with a small connector plugged into the back of the drive, usually to one side of the drive. This is the CD audio cable. If this cable is present, you need to pull this out as well. There is usually a small catch on the cable connector which you have to press in order to be able to pull it out from the drive.
Remove the screws holding the DVD writer to the drive bay (there are usually 4 screws, two on each side). Some drive bays like those in systems from manufacturers such as Dell or IBM, tend to have their drives secured using a sort of sliding rail system instead of screws. For these, there is usually a catch you need to poke in with a screw driver, and then the drive can slide out of the bay on its rails.
Slide the drive out through the hole in the front of your casing.
That takes care of taking the drive out. Connecting it to another system involves plugging in the connectors describe in 3 but is slightly more complicated because of something called a jumper setting.
Unfortunately, this can be rather daunting if you have never mucked about with the insides of a system. What I would suggest is if you are not comfortable doing this yourself, ask a mate who is familiar with this sort of thing and ask him to do it for you. Explain to him that you want to remove the drive and test it on another system to confirm if the drive is at fault. It should take about 30min tops to pull out the drive and reconnect it to another system, so it isn't that big a deal. When he is actually doing the job though, watch what he does and learn from that. My descriptions can't compare to actually seeing it done yourself. The next time you have to do something like this, you'll have a better idea of what needs to be done and you might be able to do it yourself.
Might cost you a can of beer or such for his services, but well worth the learning experience.