As was said in the other answers, the part in contact with the soil is the "foundation", or perhaps "footing" (a specific type of foundation, but there are also piles, drilled shafts, rafts - all of these are foundations).
The soil under a footing will usually be called either the "bearing surface", or "bearing strata". However, this would not apply in the case of a deep foundation such as a pile foundation unless you are specifically talking about end-bearing piles.
Other terminology you may hear and find useful in different situations:
- soil matrix, or soil profile
This refers to the layers of soils at some location, such as you will see in a boring log.
This is a wildly variable term that very often means different things depending on who says it. When a structural engineer asks "What is the bearing capacity?", they are almost always asking for an ALLOWABLE bearing capacity based on some limitation for settlement (often "1 inch or less"). However, bearing capacity also means the amount of stress a soil can experience before undergoing some kind of shear failure. These are related, but totally different things. Always be sure you understand what is being talked about when people start throwing around "bearing capacity".