Meaning:

You use this proverb to warn someone not to plan anything that depends on a good thing you expect to happen in the future, and tell him/her to wait until it really happens.
It is used to tell people not to be too sure that something good you hope for will really happen. It might not happen after all.

It is often shortened to "Don't count your chickens."
The "before they hatch" part can be substituted by "before they are hatched" or "before they've hatched."

Example:

A: Here's the list of the things I'm going to buy next month when I get money.
B: It looks great, but don't count your chickens before they hatch.

You might think the new president will fix all the problems in our company and we will get a raise, but don't count your chickens (before they are hatched).

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