I'm going to answer your questions fairly broadly as there are a lot of variations of mixers and CDJs on the market, but generally speaking, for a basic two channel mixer with 2 CDJs connected (one on each channel of the mixer):
1) A CDJ is used to play music tracks whereas a mixer accepts the raw audio signals from one of more players (CDJ's) and allows you (the DJ) to manipulate the music from one or more CDJs and control how the music is delivered to the Master out (speakers).
3) (it makes sense to answer Q3 first)... To expand on the above, a CDJ is designed to:
Provide an unaltered line level audio signal to be received by a mixer.
Cue a track (set the starting position of a track so when the play button is pressed, the track begins playing from the cue'd position, e.g. the first beat)
Skip through a track
Change between tracks
Adjust the speed (BPM) at which a track is played using the pitch fader
Make finer adjustments to the current position of a track using the outer ring of the platter (for beatmatching)
-Make adjustments to the position of a track using the top surface of the platter (usually when cueing a track)
Scratch a track using the top surface of the platter (if you're clever :) )
Create a loop on a track using the loop in/out buttons
Exit a loop using the loop exit button
There are many more functions depending on which CDJ you decide to get but these are the standard and IMO bare minimum functions of a CDJ.
Again depending on the CDJ model, tracks may be played by the CDJ from sources such as: USB stick, USB hard drive, SD card, Audio CD, Data CD, Data DVD, laptop via rekordbox link, laptop via USB connection using digital DJ software, iPhone/iPod/iPad via USB and there are probably some that I've missed.
2) Functions of a mixer:
-A mixer outputs a line level signal - it is not designed to output sound to passive speakers, it must first be amplified or powered speakers must be used.
-A mixer accepts the line level audio signal from a CDJ
-Allows the DJ to control what (if any) of the music being played by the CDJ is able to be heard on the master speakers
-The channel faders are used to adjust the volume of sound from each CDJ which is to be sent to the master output.
-The 'Low', 'Mid' and 'Hi' knobs (generally located above each channel fader) are used to adjust the frequency of sound that will be sent to the speakers via the channel fader.
-The 'Trim' knob (above each channel fader) is used to make an initial adjustment to the volume (gain) of a track before the sound is sent to channel fader. This is used when two tracks have been mastered at different volumes - you use the sound level LED's above each channel fader (green/orange/red) to determine how much gain you need to apply to a track to ensure that when you transition from one track to a next, the master volume doesn't change dramatically.
-The 'Cross Fader' can be used to adjust the volume level of two channels simultaneously. The channel faders must be up for this to be effective. Pushed to the far left, only channel 1 can be heard, in the ctr position both channels can be heard and pushed to the far right only channel two can be heard. The other positions in between adjust the volumes progressively toward one channel and away from the other. The cross fader is also used for scratching; generally one channel is being "scratched" while the other channel is playing. Youtube is your friend :)
-The "Cue" buttons (one for each channel fader) allow any sound being sent to that channel by a CDJ to be heard through the headphones, irrespective of the position of the channel faders. These buttons are used when you're cueing a track on a CDJ and when beatmatching prior to mixing two playing tracks together with the master speakers.
-There are also effects that can be applied to individual channels such as high pass filter, low pass filter, crush, echo, reverb etc. This list grows depending on how much money you spend :)
Finally, youtube is your friend :) There are heaps of tutorials which will help you to get a better understanding of what all the knobs and buttons do, including official videos from Pioneer.
I hope I've answered your questions and feel free to ask more! Also, don't forget to have fun.