A cat chasing a mouse is the classic "predator hunting prey" but, hey!, it can be rat chasing a cat for love and affection and getting it! See the vid:
It is all about socialization. If the rat and cat are raised together they don't fight, they love. Now, I wonder if we can translate that to the human race? Nah........
The problem is I, personally, cannot show a video of a cat chasing a mouse because, although it is the most natural thing in the world, I feel bad for the mouse and in any case it is a kind of voyeurism. Sure, cats chase mice but we don't have to gloat over it and put it to music. That, though, is just me and most people think different to me!....update: I have included a photo of a cat that chased and caught a mouse.It is Furby...
So here the classic cartoon:
In the real world a cat chasing a mouse can look pretty gruesome because the cat seems to want to play with the mouse, prolonging the chase. This is because cats don't get much chance, usually, to exercise their hunting skills so when the chance does come along they prolong it. And secondly, they are playing safe, avoiding a bite and preferring to batter the mouse to death with their paws at a distance. See Domestic Cat Hunting.
Above photo above by miniature. Next up we have a cute little video of a cat that is very interested in a mouse of a different kind. The pointer produced and moved by a computer mouse. This shows how easy it is to entertain our cat and we should...all the time. They get bored you know.