It depends on your
hardware. With a regular 'ol soundcard, you can record a left-track and a right track. That's it.
Of course, you can use your mixer to mix live, and record all of the mixer-inputs to a stereo track.
if you have a multitrack interface (
example), you can recurd multiple tracks at once. (Such a devide will usually come with software.)
And mix them during recording with my mixer board?
Maybe.... But generally,
no! You usually mix digitally-recorded tracks in
software. Typically, you'd use DAW (Digital-Audio-Workstation) software, rather than a simple audio editor like Audacity. You can get a "control surface" that looks-like and works-like a mixer but is actually interfacing with your DAW.
If your mixer has a USB output, Most "analog" mixers only send the mixed left & right channels through the USB port... In general, they are not capable of multitrack recording.