Running and fast walking burn calories, as well as boxing (if you want to buy a boxing bag) which will also tone your arms and upper body. Skipping with a skipping rope is good and gets your heart rate up. Swimming is also good if you have access to a pool, as well as treading water (seeing how long you can stay upright in the pool without touching the bottom). If you want to burn calories/loose weight, start by finding out how many calories you are eating each day, then you'll know what exercises to do and if you need to eat less/different foods too.
Any kind of cardio that gets your moving and your heart beat up is going to burn calories. Some things you can do around the house:
Jogging up and down stairs.
Jogging or walking around the block.
Doing yard work.