Dang, this is one of those things that would be much easier to demonstrate than to try and explain in text only. I agree that Massive Master gives a darn good start. Let's see if I can supplement what he said without confusing things!
(And I intend to seriously over-simplify here!)
Buses tend to be a way of grouping two or more individual channels together just to make mixing easier. Typically, it would be several related channels (Massive's example of drums is a good one but it might also be backing vocals, or maybe the whole brass section or whatever) with the Bus then fed to the main mix. This lets you change the level of the drums or backing vocals or brass or whatever in the mix with a single fader while maintaining your carefully set
balance within the group. Without a bus you might have to move six or eight faders at the same time and keep the level relationship the same as you do it. Obviously, there are other potential uses (for example, in live work if I don't have a speaker management system I've sent anything with lots of bass to the subs and other things to the full range speakers (using bus outputs rather than feeding it the main outs) but the first example is by far the most common.
Aux Sends are a way of pulling the signal off one or more individual channels to do something separate from the main mix. They can do lots of things--feed a channel to an effects unit and be able to mix the reverb (or whatever) in and out, create monitor mixes, either for headphones in the studio or wedges when live and so on. Another use I've had in live situations was to create separate mixes for recording and things like backstage speaker feeds.
It's worth saying that there are two sub classes of Aux send: Pre Fader and Post Fader. Some mixers let you switch Auxes between the two types while other just give you a few of each. Pre fade auxes are useful for monitoring or recording feeds because the mix on the aux is pulled off before the fader and doesn't change when you adjust the main mix. Post fade auxes are good for things like reverb effects because the effect can fade in and out as you raise and lower the channel(s).
As you can see, there can be overlap between buses and auxes--sometimes you have to take a decision which suits you best. Just to confuse you more, many mixers allow you to have aux sends on bus faders--so you can group all those backing vocalists on a bus then add the reverb to all of them at once on the bus instead of controlling a bunch of aux sends separately.
An insert is something on an individual channel that lets you interrupt the signal path, send it to some kind of processing like an effects unit or a compressor then bring it back into that channel permanently. Usually inserts are taken out immediately after the channel pre amp but some boards let you go pre or post EQ. In either case, the effect is already added before the signal gets to the fader and the only control you have is on the effects unit itself, not the mixer.
I won't try to better Massive's sidechain explanation!