Extending a concrete foundation requires a blend of strength and precision. It's a lot more complicated than fastening a couple of Lego blocks. If the extension is for a building addition, it will require an architect. An extension for a porch or deck can be handled by a homeowner, with some help. It will require a lot of digging and a rented concrete mixer or delivery by a concrete truck, depending on the size of the extension.
Dig down at the end of the existing foundation to determine the depth and width of the footings under it; they will be wider than the foundation wall. Dig deep enough to see the depth of a gravel base under the footing. Measure and mark the area for the new foundation with stakes and builder's twine, to the width of the existing footings. Dig a trench of that width and depth, including the gravel base, for the new foundation wall.
Fill the footing trench with gravel up to the depth of the existing foundation footing. Compact it firmly with a hand tamper. Fill that trench with concrete, up to the top of the existing footing and level it with a trowel. Set upright reinforcing bars in that concrete 18 to 24 inches apart up to the height of the foundation wall. Let that footing set for at least a day.
Build foundation forms with 2-by-6-inch (or wider) boards to the height and width of the existing foundation. The width of the form boards will depend on the height of the foundation; you can use 2-by-8-inch or multiple 2-by-6s. Secure those forms in place with stakes in the ground outside. If the gap between ground and form is wide, nail 1-by-2-inch spacer boards from stakes to the tops of the forms. If using multiple boards, nail 1-by-2 supports on the outside of the form. Forms must be square and solid enough to hold the concrete.
Drill holes every 6 inches or so into the end of the existing foundation with a masonry bit the size of the reinforcing bar. Drill as deep into the foundation as you can. Secure reinforcing bars in the holes with mortar, extending about a foot into the new foundation area.
Pour concrete into the forms, covering both upright and connecting reinforcing bars thoroughly. Use a board to push the concrete firmly into the forms and up against the old foundation. Level the top even with the old foundation and smooth it with a trowel. Use a long board and level to make sure the foundation top is level all the way. Let the concrete cure for a day or two, then remove the forms and backfill with dirt around the new foundation walls.