I know that the cats can see in the dark, but I have been wondering: can cats see as well in the dark as they do in the light?
Or do they see better in the light?
Cats can't literally "see in the dark", although they can see better in dim light than people can, because their pupils can expand to let more light in. So like us, they need some light, the more the better up to a point. (Glaring lights can make it as difficult to see as no light at all.)
A cat's vision is different than ours in other ways. Their colour vision is limited, and they are nearsighted. However, they are very quick to spot movement!
Great answer but I will add a few things.
Cats are crepuscular rather than nocturnal, which means they their eyes are evolved to operate at twilight. Cats have several visual adaptations to operate better at twilight.
Cats have 6 to 8 times more rod cells in their eyes. Rod cells distinguish the shapes, edges and textures rather than the colour. As a trade-off they have 10 times less cone cells than humans, which limits their ability to see colour significantly.
Cats have larger pupils than humans, allowing more light to come in. However, this in return causes cats to dazzle easier in bright light. Most crepuscular animals have this problem and many road kills are due to animals being dazzled due to car lights. As a result, cats have less vision than humans in bright light.
Cats eyesight is optimised for a few meters. Cats are practically blind at distances less than 10 cm. You probably noticed cats trying to find a treat on the ground but repeatedly fail.
Cats have poor far sight. A cat can see details at 10 m in the same way than a human sees those details at 100 m. They compensate this with their excellent auditory senses.
Finally, cats operate with their whiskers as well as their eyes. At dark, they like to be closer to a wall or fence to feel it constantly with their whiskers. They are much better judges of distances than humans and they learn the layout of a room faster than humans. Therefore, cats can operate with relative ease in pitch black compared to humans. However, this doesn't mean they see at dark.