The purpose of a lift is to be able to reach high objects and manipulate them. You might think "Why not just build a high structure?". But the main reason that we use lifts is because there is an 18 inch size limit on VEX robots in competitions where at the start of each match, the robot has to fit in an 18 in x 18 in x 18 in cube. There is no rule on expanding higher after the start of the match and so a lift allows for a much higher robot without breaking the size limit.
"Scissor lifts are assembled by creating a pivot point at the midsection of two structural pieces of metal which are crossed. Typically, the end of one piece of metal is fixed to a pivot point on the chassis and the end of the other piece of metal can slide towards the first end across the chassis. This closes the scissor, raising up the two pieces.
These lifts are nearly always assembled as pairs to equalize the forces on the lift. There is usually a platform on the top of the lift which is attached in the same manner as the bottom.
Scissor lifts may require extensive cross supports to provide stability.
The more scissor sections assembled on top of each other, the higher the lift can reach. However, it takes more torque to lift them and it becomes more difficult to stabilize the lift. The lower a scissor lift compresses, the more difficult it is to raise it.
Due to the many linkages necessary within a scissor lift and the variety of forces acting on the lift, it can be among the most difficult of the manipulators to assemble as an effective functioning system." - VEX
Scissor lifts are not hard to build the structure of but it is one of the hardest to maintain. This is due to as VEX said, the variety of forces acting on the lift. Although these lifts are seen in many use cases outside of VEX, it is not recommended unless there is a specific need for it. It is often slow due to the many linkages causing friction and is time consuming to build and maintain. It also risks the parts breaking if the balance is not right. It also usually consumes lots of space. Though there are some positive aspects as well, such as being able to lift higher than many other lifts and the stability it provides when staying lifted.
"Rack Gears are mounted to the Linear Slide Track. A Rack Bracket is attached to an Acetal Rack Truck. Then a motor is assembled in the Rack Gearbox Bracket with a spur gear on its shaft. This will allow the rack bracket assembly to travel up and down the linear slide track as the spur gear drives over the Rack Gear.
The Linear Slide Tracks are usually attached to the chassis and a manipulator, additional rack track, or platform can be attached to the Rack Bracket." - VEX
Linear Slides are lifts that could be used for simple mechanisms that does not require as much height or torque. This is due to the lift being based on gears and racks which means that the movement solely relies on two plastic gear teeth meeting each other. This is unreliable as the gear teeth could erode over time.
"These lifts are nearly always assembled as pairs to equalize the forces on the lift. Chain lifts are assembled by nesting pairs C-Channel together and attaching one set of C-Channel to the chassis with a sprocket and chain system between the pair of C-Channels. The next pair of C-Channels are attached to the chain and sprocket system. As one or more motors spin the chain and sprockets, the second pair of C-Channels slide up the first pair.
This assembly can be repeated to reach extreme heights. Spacers are necessary to keep the C-Channel sections nested together and allow them to slide up and down one another." - VEX
Chain lifts are able to lift high heights while consuming minimal space unlike the other lifts. This allows them to be used in robots where other mechanisms take up most of the space, and a compact lift is needed. You will need to check and maintain the steadiness of the chain as a loose chain will cause the lift to not function while a tight chain will snap off easily when there is tension. If only one chain lift is built, it is possible that the lift will sway and be unstable when high as the chain lift is based on just one line of construction.
This mechanism was introduced as an arm but many teams also uses it as a lift. This is due to the stability that it provides when built properly and the ability to extend very high when built on a large scale. A double reverse 4 bar lift is very common in many competitions and many teams put arms on top of the double reverse 4 bar to manipulate objects on top at a high height.
This mechanism is useful when the competition includes having to stack an object. This lift is used by pushing one object onto another from the bottom up so that it forms a "stack". By doing this, it is possible to manipulate the stacked object from the top and bottom. This was used in the 2019-2020 game Tower Takeover game quite frequently and is a little difficult to apply in other games depending on the manipulated object.
Many of the arms introduced in the robot arms section of this page could also be used as a lift depending on how high you would want the lift to be able to reach.
This table shows the pros and cons of each lift. Although some criteria such as "Fast to lift" or "Stable" could differ depending on the build quality of the mechanism.