I bought a new mic, audio-technica AT2020 but it does not work with audacity. 
The mic is connected with an xlr cable and then into the back of my computer with an adapter from a 6.3 mm jack to 3.5mm.
Obviously, I am doing something wrong here and would be extremely happy for any suggestions.
Thank you.
That mic requires 48V DC phantom power so you need a pre-amplifier for the mic and to connect the pre-amplifier to the computer line-in input if it has one (the mic input will not work).
Or you need to connect the mic to a USB interface (or to an XLR to USB adaptor like Shure X2u) and then connect the interface or adaptor to a spare USB port on the computer.
See http://www.audio-technica.com/cms/wired_mics/a0933a662b5ed0e2/ .
Gale
Thank you Gale.
This is more complicated than I thought.
I had a look at the link you sent me and yes, the Shure X2U is also mentioned there.
If I were to get an XLR to USB Shure X2U, I can connect this one to a free USB port on my computer. Meaning that the Shure X2U has phantom power integrated?
Monika
Yes X2u provides the required phantom power - see http://www.shure.com/americas/products/accessories/microphones/microphone-problem-solvers/x2u-xlr-to-usb-signal-adapter .
Also you can plug headphones into the X2u and monitor yourself without any latency, and if you do overdubs, mix in however much you want to hear of the computer playback.
Gale
I also looked at the Blue Icicle. This seems to be a simpler version. Do you have some experience with that one too?
I live in Switzerland and it is rather tricky to get good equipment here. Having shipped it into CH always costs a fortune. So, I try to avoid this if I can 
Monika
Another idea. If I get something like an M-Audio Fast Track, could I then connect my 6.5mm jack into this and connect the M-Audio into my computer?
Thank you so much.
Monika
The Blue Icicle does not appear to have a headphones output so you will only be able to monitor yourself using software playthrough in Audacity and you will therefore hear yourself late. If you don’t record your singing over a music track and don’t want to hear yourself in headphones while recording, then Blue Icicle would do.
If you want to overdub or monitor yourself then you’ll need X2u or indeed Fast Track.
I don’t personally use any of these, I am just telling you what they do. 
Gale