You may not be able to train cats the way you train dogs or they may not perform your trick on command, but mind you, cats understand what you want them to do, they just decide not to do it if the activity does not benefit them.
Cats are sophisticated human companions compared to dogs or other domesticated animals. Unlike dogs that hunt in packs and are led by the alpha of the group, cats are solitary hunters that work alone and are motivated by their survival instinct. They would spend hours observing their prey, studying their every move, gauge size, measure distance and even calculate wind direction. They would map out the area where the prey is located and plot an attack that would yield a perfect snatch. Domesticated cats often display such behavior when they are perching by the window, spending hours studying an outside bird or a running squirrel. Sometimes you may find your cat playfully attack your leg from behind, catching you off guard. They move in a stealth fashion so as to keep them from being noticed by their "prey". They maneuver their body to a form of art that makes them one of the greatest hunters in the animal kingdom.
Cats are excellent learners. They learn by mimicking others with their incredible mental and physical abilities. Kittens learn to use the litter box and groom themselves by watching their mother. Cats can also mirror their owners by copying what they do, such as opening a door or window, raiding a refrigerator or even flushing a toilet. By trial, error and imitation, cats can figure out how to perform a trick on their own.
Compare to dogs, cats' memory span generally last for 16 hours as oppose to dogs' which retains their memory for only 5 minutes. Research has found that a cat's memory is comparable to a 2 - 3 year old child and they are able to perform complex problem solving tasks such as puzzles, mazes and so forth.
Though cats may not have the most impressive short-term memory, their long-term memory is truly exceptional. Cats that have developed a close relationship with their human companions display strong affection and emotional ties to those people and can remember them for a long time. This explains why even after being separated from their long-term owners for a while, when they are reunited, cats can still recognize the person from their past. However, the emotional lingering also causes them distress when they are no longer with someone that they have been spending a long time with. Their behavior may change due to stress and anxiety of losing the close companion. This behavior is especially apparent in older cats.
Cats make emotional mapping based on the memories they have in their past. A cat that has been abused by humans, are distrustful of them. Once the memory is solidified, it instructs the brain to associate humans as threat or danger. This is why in order to gain trust from an abused or neglected cat, a slow, steady process is needed.