Houses may have spread footings that support the perimeter walls. These footings are wide pads that are continuous around the perimeter of the house. In some cases, the pads may be widened and/or thickened to accommodate concentrated loads from building components, such as fireplaces or pilasters.
A pilaster is a thickening of a foundation wall. It may be thickened to receive the concentrated load of a beam resting on top of the pilaster, or it may be acting as a stiffener to prevent the foundation wall from bowing inward.
Pad footings are similar to continuous footings except they are usually in a square, with the column or pier sitting in the middle of the square. It is common for houses to have strip footings around the perimeter and pad footings on the building interior under columns.
Piles are typically used instead of footings where the soil quality is poor. They are, generally speaking, more expensive to install and have to be driven into the ground with specialized equipment. They can work one of two ways. First, piles can be driven down to a point where they bear on bedrock or other sound substrate. Or, piles can be driven into the soil far enough that the friction of the soil against the sides of the pile is enough to resist downward movement.
Incidentally, if a house is supported on piles, they probably won’t be visible and it may be impossible to know it.
Piers are columns that may be completely concealed in the soil or may project above it. Think about the piers that are commonly used to build exterior wood decks and porches. These piers may be poured concrete, often with the concrete poured into a cardboard cylinder in a hole dug in the ground. Piers usually, but not always, have footings. Piers can either be thought of as posts or columns, or could be thought of as short piles that bear on their ends.
Grade beams are usually concrete beams that are supported on footings, piles, or piers, and are located at grade. In some cases, they extend below grade; usually they extend only slightly above grade. Grade beams transfer the loads from the building down to the footings or piles.
Caissons are foundation systems created by drilling holes and filling them with concrete. A caisson pile is a cast-in-place pile that has a hollow tube driven into the ground. The earth is excavated from the tube, and concrete is poured into the tube. Some caisson piles re flared out at the bottom to create a larger bearing surface. These are sometimes called bell caissons.
This has been a mostly technical discussion- but the important take away is that footings and foundations are important to the stability of the house, expensive, and mostly out of sight.
Footings and foundations should be strong so they can transfer loads and durable with respect to exposure from air, water, soil, and insect attack. Most modern footings are concrete (sometimes reinforced). Footings on older buildings may be brick or stone.