What type of insulation is needed for exterior walls on a brick house?
If you wish to preserve the brick exterior, you have the following options, depending on the wall construction:


If you have a structural brick wall (e.g. multiple-wythe brick or brick veneer on block) then you will have to insulate the wall from the inside. This will entail applying rigid insulation board--either foam or mineral wool--over the inside of the wall, and then covering that up with drywall.
If you have a brick cavity wall (e.g. brick veneer on the outside and structural brick or block on the inside) then you can blow water-permeable insulation like perlite into the cavity. You could additionally insulate this wall from the inside as per above.
If you have a brick veneer on wood framed wall, then you'll need to insulate the stud cavities normally. You can use fiberglass batts, mineral wool batts, closed-cell spray foam, or dense-packed cellulose. All will be effective. One advantage to dense-packed cellulose is that you don't actually need to take down the drywall; you just drill holes in the top and blow in the cellulose. This could save you a ton of money if you haven't already removed the drywall. If you have to or plan to remove and replace the drywall anyway, my personal recommendation would be installing either mineral wool batts or dense-packed cellulose in the stud bays, covering the open stud bays with an additional layer of rigid mineral wool board, and then applying the drywall over that. An added bonus is that this will all but completely soundproof the house.