I'll break this into short, hopefully digestable topics.
Linked suspensions, and why they have been developed.
As vehicle manufacturers strive for improved handling, smoother rides, and longer component life, they realized that they needed a suspension that would offer better control of axle movement during suspension cycling and cornering.
Linked suspensions (typically known as "coil suspensions" due to the spring design), refer to the way the axle is located under the vehicle. Through a series of "links" (commonly called control arms), the axle can be located front to back, side to side, and radially with regards to the axis of rotation of the axle shafts. Using different materials, link mounting points and link lengths, one can very precisely control the movement of the axle to obtain precise and predictable handling characteristics.
Without going into design details of suspensions (which is outside the scope of this website), here are some of the effects of installing lift kits on vehicles with linked suspensions.
Link angle, relative to the ground. As the link angle increases (due to a lift kit being installed), there are a couple things that happen to the suspension (ignoring the steering geometry changes).
1) Road noise and vibration are transferred to the frame at a greater angle than a non-lifted vehicle. Vibration travels strait up the link, and when it hits the frame, it will is transferred to the frame....the angle at which this happens will greatly affect how much you feel. Vibration transferred to the frame at a low angle causea the frame to vibrate end to end, vibration transferred perpendicular to the frame will will cause the frame to vibrate up and down (as opposed to end to end). Due to the relative rigidity of the frame (end to end vs up and down) means you will feel that vibration more and more as the link angle moves away from parallel to the frame and closer to perpendicular to the frame.
2) Axle movement. As the links move lower in their arc of travel, you get move horizontal movement per unit of vertical movement. This means that when you hit a bump, your axle can't move strait up, it will move in a arc, forward and up at the same time. So, as you hit that bump, not only is your vehicle moving forward in relationship to the road, but the axle is now also moving forward in relationship to the vehicle. This increases the vibration transferred through the suspension and into the frame. It's basically a double wammy.....you have increased vibration intensity moving up the link, and you are changing the angle at which these vibrations are transferred into the frame, resulting in a increased "feel" of the vibration.
***oops, gotta run, I'll continue with this later......
Ok, it's later.
3) This is a special case of #2. As I already pointed out, after a lift, the links are lower in their arc of travel. Now, if you have ever compared lengths of upper and lower links, you will notice that upper links are a little shorter than the lowers. This allows the castor angle to remain constant as the suspension cycles (more consistant handling). Basically, as the suspension cycles, the castor will remain close to the same angle, despite the change in the angle of the arms relative to the road. This works great as long as both sides of the axle are moving in the same direction. But what happens when you put one tire on top of a big rock, and one tire is still down on the trail? The side that is "up" will try to change the pinion angle one direction, and the side that is "down" will try to change the pinion angle the other direction....so in effect, you are trying to twist your axle housing. With a stock height vehicle, this twisting is minimal, because all of the arms are pretty close to being parallel with the ground (and frame)....this means that you get very little horizontal movement relative to the vertical movement. But when you lift the vehicle, this changes. Now you have a much larger difference in where the links are trying to position the axle (due to the links being much lower in their arc).
4)Soft bushings usually allow adequate movement (to compensate for issue #3), but be careful. Soft bushings wear quickly, and do not allow as precise of control of the axle as compared to harder bushings. The downside of harder bushings is increased road noise being transferred up the links, and additional stress being placed on the links and link pockets (mounts). This last bit of stress mentioned is created by a misalignment between each end of the link. As your vehicle sits on the road, the bolts running through the bushings on each end of the links are parallel to each other. As one side of your axle lifts up , the bushing on the axle end of the link stays parallel to the axle, while the frame end stays parallel to the frame...but the axle and frame do not stay parallel to each other....basically trying to twist the link. So, again, here hard bushings are not so good. A well designed lift kit will make an allowance for this twisting by either making a 2 piece link that will twist around itself (like Tera), or by using a spherical joint on one end or the other (like Rubicon Express, Currie, Full Traction, Fabtech, etc). This is definately something you want in a lift kit. Interestingly enough, OEM's deal with this in a different way. They design their links to twist, which is why they are almost always stamped pieces, and never boxed.
Now, all of the above applies to 4 link suspensions....and most of it applies to 3-link suspensions as well. The difference here is that on a 3-link, one of the upper links is not used. This eliminates the issue of the suspension trying to twist the axle housing in opposite directions. You need 2 attachment points to twist the housing in either direction....remove one of the 4 points, and it will only twist one direction at a time. The downside is that if the single upper arm fails, your stuck until it is repaired.
I have not mentioned track bars (AKA panhard bars) either. These are usually not counted when counting "links". There are a couple things to watch out for though....as you lift your vehicle, the track bars will need to be lengthened to keep your axle centered under the vehicle. If you don't do this, your axle will be offset to one side, which can contribute to tires meeting sheet metal when you stuff a tire into the wheel well. Some people simply move the axle end of the track bar closer to the drivers side of the vehicle to compensate for the longer length between the mounts....the problem here is that you are now moving the pivot point away from the pivot point of your steering system, and can induce bump steer. It is better to have a proper length track bar. One last thing to remember, is that the bar will now be at a steeper angle at each of the joints, so be sure that the joint can operate in the new range.
Ok, 2 more things.....I'll keep them short, because most of you have already skipped over this post anyway......
A special case of the "3-link" is becoming very popular on the rear axle. A "triangluated 3-link" is just what it sounds like. 2 traditional lower links, but a triangle shaped upper link. This link has an attachment point on each side of the frame (driver and passenger), and one single attachment point in the center of the rear axle. This design allows the upper link to keep the axle centered underneath the vehicle, making the track bar useless...so you can get rid of it. It is a very functional design. But you will rarely see it on front axles because of the offset differential moves teh front driveshaft off to one side, and the transmission and engine oil pan get in the way.
And finally...WHY would someone want to deal with all the problems of a linked suspension? predictable handling, no axle wrap, the ability to tailor the anti-squat number to the application, easier to tune for specific applications, and generally a better ride quality. The only real downsides are the complications when setting them up (which is one of the reasons quality lift kits are quite expensive), and when the link angles get too extreme, the suspension has a tendancy to do some quirky things...most of which must be experienced to be fully understood.