Fence posts break for all kinds of reasons. Over time many will rot and simply break off at ground level or a windstorm will take down a perfectly good fence. I even know a fellow who had a moose literally walk right through his fence, breaking a post off and taking down two sections of his fence. However it happens, once a fence post is broken, replacing it can be significant job, often involving expensive or specialized equipment. However, here's an idea that works on lots of fence posts and might save you some money and a lot of aggravation.
Start by freeing the fence post
Assuming your fence is still standing but your post is broken at or below ground level, your first task is to detach the fence from the post. If the fencing is toe nailed into the post, use a pry bar or a nail puller to back the nails out and free up the post. If the fencing is attached to the post with fence clips a reciprocating saw is probably the easiest way to cut right through the nails and free the post up.
In many cases you only need to free up the fence sections on each side of the post, then swing them back a foot or so to give yourself enough room to work. If that's not possible you'll need to totally remove the sections of fence on each side of the post for access to the broken post.
Pulling the post
Dig down around the post to loosen the soil around the base (and expose the stump if the post is broken at or below ground level). Wiggle the stump back and forth a little to help break the hold of the ground. If you're lucky, at this point some fence posts that have been poured into concrete will actually lift right out of the ground. Unfortunately most of the time this isn’t going to happen, but it's worth a try.
If your post is still firmly in the ground, make a loop in your steel cable fastening it through a U-bolt then around the post 'stump' and pull it tight. Stand the jack up close to the hole and attach the U-bolt on the other end of the cable to the jack itself.
Now simply operate the jack to pull the post 'stump' out of the hole. After jacking the lever a few times stop and loosen off the cable and reposition it lower down on the stump, then continue jacking.
When the post comes out of the hole, you'll be left with a perfect sized hole, just right for your new fence post. All that's left is to drop in your new fence post, brace it up plumb and reattach the fencing sections.
Using this technique you may be able to turn a really ugly job into a manageable project you could do half a day.
A couple of thoughts
This process obviously works best for posts that are sunk directly into the ground, however, it will work on posts are just placed into concrete as well.
If your fence post isn't rotted just broken, you may be able to use this same technique to actually pull up the post and the attached concrete footing (after digging around the footing to loosen the soil).