Do cats see in black and white like dogs do?

It's a misconception that either cats or dogs see in black and white. They can't perceive the same range of colors that we do, but they do perceive some colors.
Color vision depends on how many types of cone cells you have in your eye. Animals with four (like birds) are called tetrachromats. They see the broadest range of colors. Animals with three are called trichromats. They include humans, apes, and Old World Monkeys.
Cats and dogs, like many other placental mammals, are dichromats. They have two types of cone. That's very different from having only one type of cone (which is the case for monochromats, like whales and owl monkeys). In experiments testing color discrimination (including a new study conducted in 2016), both cats and dogs have performed in ways that suggest their perception is similar to that of a human deuteuranope. In this condition, a person can see yellow-greens and blue, but not red. Reds and oranges are perceived as being olive or yellow-green.
https://www.ncbi.nlm.nih.gov/pubmed/?term=neutral+point+testing+of+color+vision+in+the+domestic+cat