Turn mower on its side, with the carburetor on the high side. Peer in the hole where the primer bulb sits. Clean away any dirt or crud. In the hole you will see a thin snap-ring, with many feet holding it in. With a screw-driver, carefully pry out on that ring, and the bulb will come out, maybe with the ring. If not pry the bulb out, too. They are not glued in.
Then straighten out the ring, place the new bulb in the hole, and carefully push the ring down into the hole, making it snug all around. Don't puncture the new bulb!
Before installing the bulb, i always use a piece of tubing to blow thru any holes under the bulb. And, if you have the correct size of socket, it can be used to seat the ring!
HTH: Rustyj

You may just need to clean the carb. Actually, if is one of the little plastic top o tank carbs, a whole new carb is under $20

Use a 3/4", 12 point deep socket to seat the ring. Tap lightly with a hammer.

If you want to remove the bulb and keep your sanity, you will need Briggs tool 19461. Is the bulb cracked or damaged? The primer circuit is made up of a flapper valve in the carb gasket, the primer bulb, and a check ball behind the bulb. The problem could be in any of the three. Get the model, type, and code off the engine and take tomplum's advise and get a whole new carb.