Examine the wall you plan to build against to make sure it is solid, whether using full brick or thin brick. Decide on a pattern and type of brick; full brick is usually laid with a 2-inch edge out, while thin brick may have either a 2-inch or 4-inch face. Measure the wall with a tape measure to determine square footage to buy brick; the coverage will vary with the type of brick and pattern of installation. Cover the floor around the wall with drop cloths to protect flooring.
Prepare the wall. Staple waterproof membrane over the wall with a construction stapler and cover it with metal lath, nailed to the wall with shingle nails and a hammer for full brick or mortared thin brick. Make sure the wall is clean and sound for thin brick to be applied with mastic like thin-set mortar. Snap level chalk lines the width of the wall to guide installation, spaced according to the pattern selected.
Spread a thin "scratch coat" of mortar over the metal lath with a flat mason's trowel; rough up the surface horizontally with a stiff bristle broom or leaf rake for full brick or mortared thin brick walls. Let the scratch coat dry 24 hours. Leave the wall bare and dry for thin brick mastic.
Use a flat notched or pointed mason's trowel to spread mortar or thin-set mastic on the wall and on the base for full brick. Scratch the surface of thin-set horizontally with the notched edge of the trowel. Lay the bricks in place, pressing them firmly into the mortar or thin-set. Put mortar on the end and top joints of full bricks; leave 3/8-inch gaps on thin bricks to be filled later with grout.
Install bricks over the wall, following the chalk lines and checking every course with a level. Finish the joints on a full brick wall every course; use a steel finishing tool with flat and convex ends to press the joint mortar firmly into place in either a square or concave shape. Let mortar or mastic set on thin brick walls, then use a grout bag, a cloth bag with a pointed metal tip, to push grout into the brick joints. Wipe excess mortar or grout off the bricks with a damp sponge.