Removing a fence post with concrete attached requires some hard work and may even seem impossible if you aren't sure what you're doing. Unless you are willing to spend money to hire a professional or pay for a load-lifting machine, you will have to dig the posts out yourself--concrete and all. There are steps you can take to make removal easier and save yourself some unwanted frustration.
Use a posthole digger to dig away dirt around the post and concrete.
If the ground feels solid and the dig is difficult, pour water around the post. This will loosen stubborn soil and soften your dig.
Shake the post. If it wiggles freely, you are ready to pull the post out of the ground. If it still feels solid, then keep digging. Most posts are two to three feet into the ground. You could also bang on the sides of the post with a hammer to loosen it, but be careful not to crack the post.
Simply lift the post out of the ground with the concrete attached. If it's too heavy, have someone help you. You could also use the pick end of a tamp-bar (a solid steel bar used for fence installation) to crack the concrete off the post. Drive the bar between the post and concrete with a hammer and the concrete will crack. You can then lift the concrete out of the hole in pieces.
If you want to keep the post, use a hammer to crack the concrete off of it.
Many are tempted to try shortcuts when pulling out posts, such as attaching a chain to the post and pulling it out with a truck. This will almost never work and will only succeed in snapping the post in half, making removal even more difficult. Only a machine with upward load lifting abilities (like a Bobcat) will be able to pull posts out without breaking them. These machines are expensive to rent, but are worth it if you really don't want to do the physical labour.
Removing a post and concrete is physically strenuous and involves heavy lifting. Do not attempt it if you're not in good physical condition.