Before you replace your mailbox post, you'll need to remove the old post. This task's difficulty depends on how you or the previous installer secured the old post inside the ground. A post set in cement is a lot more difficult to remove than a post that has only been buried. If your post's cement base is set too securely, you might need to choose a different area along the front of your yard in which to install your new mailbox.
Dig around your post with your shovel. If you hit concrete, widen your hole's circumference until it is larger than the post's concrete base.
Water the soil inside and around your hole and mailbox post to loosen the soil.
Wiggle and pull at your mailbox post. Continue to attempt to loosen it from the ground. If you make little progress, you can try to pry the post out of the ground with a 2-inch by 4-inch wood scrap and a car jack.
Screw your 2-inch by 4-inch wood piece to the mailbox post at a 90-degree angle. Attach the piece at a height just low enough to fit your jack beneath. Place your jack beneath the wood piece and pump the jack's level. Continue pumping to lift the post and its base from the ground.