Most home septic systems consist of a concrete septic tank and leach field, plus the associated piping between them.
The tank is made with baffles to prevent solids from migrating to the far end of the tank until they’ve been broken down by bacterial and mechanical degradation, and the leach field is primarily coarse gravel sloped away from the tank, which handles liquid runoff and allows the liquids to percolate into the soil beneath. All of this is covered with soil and sod as to be invisible from above.
You may also have what is called a “dry well” for gray water. Gray water is water from showers, sinks and (especially) laundry waste water. That is, it has no excrement in it, and therefore doesn’t need the same sanitary treatment. The dry well is typically a deep pit filled with the same type of coarse gravel as the leach field (in a different area, so they don’t interfere) which allows the laundry water to settle out as the soil around the dry well can accommodate it. (The smell from the dry well will also knock you down if you have to open it again years down the road. It’s much worse than opening a septic tank.)