The question implicitly refers to domestic cats. Cat guardians probably know the answer through their experience of seeing their cats bring mice into their home and release them. If alive and able mice sometimes run under furniture such as sideboards. Domestic cats will silently wait and watch an exit to the furniture in the expectation that the mice will eventually come out, whereupon they pounce and kill.
This more or less sums up how a domestic cat hunts mice in a domestic environment. Domestic cats are famous for their stealthy approach to prey. When hunting, the domestic cat can be still or move stealthily and they are always as silent as possible. This deadly behavior is combined with patience and persistence.
They use their highly developed senses of smell, sight and hearing. They select the best spot (e.g. outside a mouse nest) and wait while making careful use of available cover. The slightest rustle or the smallest movement is detected whereupon the domestic cat may freeze or stalk.
If the cat believes that the prey will move towards them they will pause and wait. If it is unlikely that the prey item will move innocently towards them they will move forward in short quick advances, while barely making a sound, interspaced with frozen pauses. Domestic cats crouch low and tense themselves. Sometimes before making a final pounce they will sway their heads from side to side in order to form an accurate 3-D picture of the position of the prey. This allows them to judge precise distance for an accurate pounce.
With mice, the final pounce is a simple downward action whereas with birds it is more complicated. This is because domestic cats anticipate that the prey will fly upwards to escape. As they close in they anticipate this and are ready to perform an upward leap swiping at the bird with both front feet at once. These actions can be seen in kittens when they play.
One aspect of domestic cat hunting is their extreme patience and persistence. They can wait for what seemed like hours at the exit of a mouse’s nest. The patience often pays off. If allowed to prey upon wild animals, the dominant pray for domestic cats are small mammals, principally rodents and occasionally rabbits or hares (if available). They attack birds but birds are harder to catch and therefore are not a priority. Insects are often chosen.
Domestic cats can reveal their intentions just before attacking by slightly twitching their tail. This may become a wagging movement in the seconds before the attacking leap or sprint. Tails are primarily balancing organs.
When an adult cat plays with a half dead mouse it is not a deliberate act of torture. Rather the cat is not hungry but is instinctively driven to hunt which drives then to pounce and pause.
Some more on hunting