Mixing usually means combining multiple audio channels with a fader or potentiometer for each channel that lets you determine the channel's sound level in the overall result.
In the simplest form, each input channel is a mono channel and the output is single channel too. You can think of each input as a pipe with a valve, and the output as a drain beneath the valves that collects all the water.
Many sound sources are typically fed to the mixer as mono sources--vocals, guitars, bass, individually miked drums on a drum kit. Other sound sources are typically recorded in stereo, such as a pair of drum kit overhead microphones, a synthesizer that delivers stereo output, samples of existing recordings, etc. So a typical mixer will usually support both mono input channels and stereo input channels. In both cases, each channel will usually feature a pan pot (panoramic potentiometer) in addition to the fader that lets you determine how much of the input that is sent to the left or right channel. This can either be used to mimic the physical placement of the instruments and vocalists (e.g. vocal and bass center, piano slightly to the right, guitar slightly to the left), or to create sound effects (e.g. doubling a vocal take and panning each take hard left and right for a richer sound).
When you mix live sound, the sound source for each channel strip will come from the stage--microphones, instruments, etc. plugged into the mixer unit, and the output will be sent to the main speakers in the venue. When you mix and produce recorded audio, the input for each channel strip will typically come from one or more multi-track recordings.
Physical mixers are usually organized as a series of vertical strips, where each strip represents an input channel. You will typically find the above mentioned fader and pan pot, as well as basic tone or equalizer controls. More advanced mixers will also let you either insert effect units or send input to external effect units. To come back to the water pipe analogy, the raw sound input enters the top of the channel strip, then passes through sound effect units and tone control, is split into stereo through the pan pot, and is finally blended into the overall mix through the channel fader.
Advanced mixers often let you group mixes on so-called busses. There are many more variations and concepts like this, but the channel strip with the fader and the pan pot are the basic concepts you need to understand when we talk about mixing audio.
Many software packages mimic the layout and terminology of physical mixer units.
Advanced physical mixers and software mixers often feature automated mixer controls. If you want to gradually pan an input channel from left to right, then you can record the movement of the controller so that it is automatically played back when the final mix is made. This way, a mix can be shaped and sculpted in very great detail by adding more and more recorded mixer control changes.
The Art of Mixing
The mixing engineer and producer usually have several goals with the mixing process. For example, if producing a hard rock song, the mix will probably feature prominent electric guitars and drums with a punchy sound. At the basic level, this means the faders with the electric guitars and the drums are pushed a little higher to make them a little louder in the overall sound picture, or conversely pushing all the other faders a little lower. At a more basic level, the mixing engineer and producer will usually try to make sure the overall sound is balanced--no one instrument or vocal drowns out the other instruments and the mix feels not too deep and rumbly or too thin and scratchy. Some of this is achieved by fading input channels up and down--e.g. increasing the bass channel will make the overall mix more bass-heavy--but tone controls and effect units are also used to shape the sound.
Even when limited to just faders, pan pot and tone controls, it's difficult to come up with hard and fast rules about mixing. The number of possible combinations and variations quickly skyrocket and the perception of a good or bad mix is highly subjective, and changes with trends in popular music all the time. Even the sound of classical recordings have changed dramatically over the decades as recording, mixing and playback equipment has improved, and not just because of the technological improvements. And then there are experiments where an artist or producer deliberately breaks away from the norm in order to achieve a certain artistic effect.
There are a number of ways you can learn more about mixing, either through audio engineering classes, apprenticeships, or through trial and error, but common to all of them is that you need to be willing to spend the time. I've found that trying to mimic the sound to be a fairly long, but ultimately rewarding process. Identify records and songs that you like, not for the song itself, but for the way it sounds. Then try to second guess how the mixing engineer and producer arrived at that sound. Some things will be fairly obvious, but a lot will be very subtle and, to a large degree, subject to your material, which of course differs from the original artists material. If you had the chance later on to learn what the producer actually did, you might find it was something entirely else. But the goal isn't to produce copies of other people's work. The point is that you'll learn a lot, and possibly develop your own style along the way.
See also General rules for panning?, How do I get a mix with two heavy bass elements to not sound muddy?, Mixing piano and guitar. For more questions and answers related to mixing, follow the mixing tag. Feel free to raise more specific questions if you can't find questions and answers that already cover any follow-up questions you might have.