You should choose easy difficulty if you have not played before.
After choosing a map click the blue arrow to buy a cat.
Click the cat you want, and then cat will follow your cursor until you click again.
When you click you will permanently place the cat (You can still sell to get rid of it). You cannot place a cat on obstacles or the path (It will turn grey).
Click a cat that is placed to upgrade it.
CONTROLS:
Space - Start round/speed up
P - Pause/open settings
C - Cancel buying a cat
S - Sell cats you click (you sell the selected cat if the upgrade menu is open)
A - Buy hero hotkey
Q through O - Each of these keys will buy a specific cat
D - Buy Molly