you don't need a lot of examples to understand the difference. a mixed record sounds cleaner than an unmixed record. that is basically it.
if you produce a beat, you have different instruments and/ or samples that make up the beat. say you have a very simple beat made up of a drum track (kicks, snares, hihats), a bass track, a piano track and a synth, plus vocals. the different drum sounds, the bass, the piano and the synth all have their own track and need to be mixed to sound good together. so the first thing you normally do is to adjust the volume of the different tracks, because you want some sounds to be louder than others.
another thing is panning, which means positioning sounds more to the right or to the left. for example, drums and bass make up the core of the track and therefore, they are normally placed dead in the center, which means neither left, nor right (but if the drums contain percussion, the can be positioned to the right or to the left). but other instruments can be placed more to the right or to the left, especially if you have a lot of different instruments, because if all of them are positioned in the center, things sound cluttered and flat. by adding reverb effects you can also move sounds more to the back ( together with the volume and the eq, basically the lower (both volume and eq) the sound and the more reverb you add, the farer away a sound appears). so with these elements, volume, reverb and panning you can assign all the different sounds a specific spot.
another thing you need to take care of are frequencies. each sound is made up of different frequencies: highs, lows and mids. if you take our example, the bass sound has a lot of low frequencies, and less highs and mids. since the piano also has low frequencies, bass and piano together will sound muddy, because your ear won't be able to really distinguish between the low frequencies of the piano and the low frequencies of the bass. so what you will to is, take an eq, and cut the low frequencies in the piano sound, to make room for the bass. then, you will probably take away some of the highs and mids in the bass, to make room for the piano. the same is true for the synth, which, if it's a lead sound, will probably consist of mainly mids and highs. in order for your ear to distinguish between the synth and the piano, you could cut highs from the piano and take away mids from the synth. then you do all of these things for the vocals, which will probably consist of different tracks too. when you're done, your song will sound cleaner than it did before, because now your ear can differentiate better between the sounds that make up your record. it's kind of like tiding up a room. if things are just cluttered all over the place it looks bad, and even if you know where everything is, it's more difficult to find what you are looking for. after cleaning it up, everything is where it's supposed to be, the room is more pleasing to the eye and if you need something, you know where to find it.
another thing you do when mixing is compression. that's a bit more difficult to explain to someone who isn't a sound engineer, but it's mainly about the dynamic and the perceived volume of sounds. since a sound that is too loud will sound distorted, you lower the loudest elements of a sound, which allows you to give more volume to the overall sound before it distorts. compression also helps you to "glue" the different sounds together.
if you want to know what an unmixed/ badly mixed song sounds like, listen to the first three mood muzik mixtapes by joe budden. than compare it to any of the songs on compton and you will know the difference between a good mix and no mix.
Feed us with some examples 
