Chisel the mortar away. Tap on the chisel with a hammer. Wear protective eye gear as you do this. There are electric grinders that can do this job much more quickly, but these devices can easily damage brick. A grinder is not to be used on vertical joints, only on horizontal ones, and they can only be used if the joint is larger than ½-inch wide. Dig out the mortar to a depth twice the width of the joint. Do not to damage the brick during this process. Cutting the brick will lead to erosion of the brick at a later date.
Around 1930, the ingredients used to make mortar changed dramatically. Old mortar was made from lime and sand, but new mortar had a concrete base that made it much harder. This new type of mortar works well for modern bricks, but when paired with old hand-made bricks, the mortar can seal in water and accelerate damage. You must choose the appropriate mortar for your home. To be certain you're using the right type of mortar, try sending a sample of your old mortar to a materials testing company to have it analyzed.
Use a spray bottle to spray a mist of water over the joints before adding the mortar, to slow the drying process. Mix the wet mortar according to the manufacturer's instructions. Insert the mortar using a a narrow tuck pointing trowel. Apply the mortar in layers, pressing the mortar in firmly as you go. You can chisel away any uneven areas after the mortar has dried for 24 hours.
Brick repointing is not difficult, but the stakes are high and there are many ways that repointing can go wrong. By using a mortar that is harder than the brick or damaging the brick itself during the chiseling process, you could accelerate the decay of your brick and incur expensive repairs down the road. If you're not ultra-confident in your ability to point your own brick wall, hire a professional.