Chip away at the joints of the flagstone, but be careful not to damage the stone or adjacent stones. When all of the mortar between the joint breaks free, use a large flat chisel to pry underneath the flagstone. Pry it out from different angles, being careful not to break or crack the flagstone. The mortar seal underneath the flagstone will easily break and you can lift out the flagstone.
Rent or purchase a diamond blade saw to cut the joints if a hammer and chisel won't easily do the job. If the flagstones are not set in mortar, you can pry them free using a crowbar.
Cut or chisel out some of the hard mortar under the flagstone, to create a surface that will allow you to apply mortar to the flagstone, and so that when complete, the stone will be level with the other stones.
Create your mortar using a pre-mixed mortar mix. Use a wheelbarrow and add water to your mix, mixing it with a shovel until you have a consistency that is firm but not slushy, one that just runs off of your trowel. Apply the mortar to the base of your flagstone, install, and reset your joints.
Clean up the flagstone of any excess mortar with water using a rag or sponge. Let the newly laid flagstone set for 30 minutes, and then clean the outside of the flagstone thoroughly. Don't walk on the flagstone for 48 hours and wait a week before you put any heavy weight or significant traffic on the walkway.