Hold on a second before you grab the biggest hammer you can find. Take a quick look at what you're dealing with.
Is the bolt bent or snapped off?
How much is sticking out?
Is the concrete around it already looking rough?
Why are you taking this out anyway?
This quick check will save you a ton of trouble later.
Here's how to get them out without wrecking your concrete:
If it's broken off or sitting flush:
Break the seal: Take a punch and give the center a good whack with a hammer
Twist it out: Grab what's left with locking pliers and work it out
Get the shell: Use a narrow punch to collapse the outer sleeve
If there's still thread showing:
Take off the nut and washer
The magic trick: Hit the top of the stud straight down - you'll feel it pop loose
Now you can just pull the whole thing out
This is where most people mess up. Once the anchor's out:
Look it over: Check for cracks or chunks missing
Clean it properly: Get every bit of dust out - use a brush and air
Now for the big question - what to do with that hole?
Choice 1: Use it as-is
Only if the hole is in perfect shape
Must use a brand new anchor
Choice 2: Go bigger
If the hole's a little rough
Drill it to the next size up
Choice 3: Start fresh
If the hole's trashed
Fill it and drill nearby

Slow and steady wins - rushing just makes more work
Clean holes hold better - no exceptions
When unsure, go bigger - better safe than sorry
For important stuff, get help - safety first