I'm very limited on time so others will have to get more in depth:
First off:
2-way = mids/tweet
3-way = midbass/midrange/tweeter
^ This is how it's usually defined. If you hear someone say 4-way, most likely they're just adding the subwoofer into the mix. But, for the most part, the subwoofer is assumed and isn't included in the "2-way vs. 3-way" talk.
Active = running non-passive crossovers. Instead, the user can control crossovers from the headunit or processor, or whatever. Most active units allow you to do t/a (time alignment), g-eq (graphic EQ), p-eq (parametric EQ), and some other things. Instead of using 2-channels of an amp with a passive inline, you now use 4 channels for 2-way and use the active crossover to do what the passive did. If you go 3-way you'll need 6 channels of amplification.
The big benefit is that due to music being so varied, sometimes you need to change the way your speakers are playing. With active you can change x-over, slopes, etc on the fly. With passives you don't have that ability (unless you want to unscrew wiring and remount it different attenuations on the passive crossover itself).
The headunit would control all this for you (or the headunit would act as a link between it and the processor, IF you buy an external processor). Not all h/u's can do this, and for one BIG reason- the tweeter/midrange x-over points. To make it simple, imagine you have a 2-way setup (mid & tweeter). Now, most h/u's can highpass a speaker at 80hz, 100hz, 120hz, etc. But, rarely have I seen one go higher than 120hz. So, ...what does this mean for the tweeter? Well, it means it's going to fry since it's not being crossed high enough. so, in this case you'll need something that can high-pass at 2 different points: one for mid, and one for tweeter. The tweeter high-pass usually occurs around 3000hz or so. Sometimes lower; depends on the tweeter.
You need a headunit capable of this, so an active one is where you go. So, now you're in this situation where you've got 2 speakers (forget sub for now). You need to make sure that each one is playing within it's safe range. Here's a basic breakdown:
Mid: plays from 80hz-2500hz. You need crossovers that support this, and you also would like to have different variations of slope.
Tweeter: Plays 2500hz-20,000hz. Same deal.
Now, that you've got that taken care of, we're back to being able to actively control these slopes, & x-overs...
Say you're listening to some 80's music (only thing I can think of). Your midbass can safely play down pretty low without much distortion. So, you set your HP for maybe, 71hz. You put a somewhat steep slope of 12db on it and call it a day.
Now, you pop in some rap. Rap is known for having extremely higher output in any bass range. Now your mids are slapping and bottoming out. You need to fix this...enter the x-over... Set the x-over to 80hz with a steep slope of about 24db. This should cut out the low frequencies from playing through your mid and save you from having to buy a new one.
The same situation goes for tweeter. Set how it needs to be and listen.
Time alignment is another important feature of SQ setups. In a nutshell it allows you to fire speakers at different times so that the sound gets to you at the same time. You set a delay on the speakers closest to you so that the sound from the left doesn't hit you before the sound from the right.
I left out EQ info because you probably already know what an equalizer does, but if you need more help, let me know.
*tired of typing*