The well known Heimlich Maneuver or 'Hug of Life' for removing obstructions in human throats works for dogs and cats too.
The objective is to compress the lower chest forcibly to raise air pressure inside the chest to blow the foreign material out of the throat or the larynx.
Here's what you need to do:

• Place the pet on its side on a hard surface.

• Place both hands behind the last rib and press down quickly and firmly - release immediately and repeat rapidly several times.

• Try to direct the force of your hands slightly forward - this will make your efforts more effective.

• Have someone carefully open the dog's or cat's mouth wide and attempt to retrieve the material from the throat as you compress the chest.
Once the airway is cleared, if the dog or cat isn't breathing, obviously artificial respiration is necessary.
Now, the important thing here to remember is - even though you've been able to release what the dog or cat is choking on, you still want to get them to a veterinarian to make sure no damage is done.
Now let me give you my version of the Hemlich Maneuver that I have used many many times on medium to larger dogs.

If you have a medium to large dog what you need to do is:

• Straddle your dog from behind.

• You put your fist just behind the last rib in the center - lifting up and forward - several quick motions.

• If there is a second person and they can open the dog's mouth - it works better.

And in many cases what is stuck in your pet's throat will come out.

Remember, then get your dog or cat straight to the veterinarian.
- Warren Eckstein