Basically, you need:
1) A microphone that'll plug into your computer
2) A program that can record from said microphone
If you're looking to just sing some songs without really caring about the production value, then Audacity (mentioned in previous post) + a $10 plastic microphone from your local electronics store will do the trick. There will probably be some noise on the line and you won't sound anything like the professionals but it'll get the job done. If you're going this route, don't spend any more than $15-$20 on a microphone; for the purposes of recording songs, it isn't worth spending more than that on something plastic.
If you're looking to do something cleaner and more professional, then you're going to have to spend some money to upgrade your microphone. The AT2020 USB costs about $100 and it plugs directly into your computer without having to get a separate interface/phantom power/cables/etc. Just to note, I've never actually tried the AT2020 USB but I have used its counterpart, the regular AT2020, to record an entire album's worth of stuff and it really is quite fantastic for the price.
there are many cover on youtube inspired me to record one... and idk how to start 
