The core is a collection of muscles which stabilize and move the spine. Close to the spine and deep inside the abdomen is the inner core, which is comprised of the diaphragm, pelvic floor, multifidi, deep cervical flexors, and transverse abdominus. These muscles engage first during movement or breathing to protect the spine.
The outer core muscles are also responsible for stabilizing and protecting the spine, but they also have more defined movement functions. The anterior muscles (abdominals) are the most well-known members of the outer-core assembly. The lats, spinal erectors, glute complex, quadratus lumborum, and hip flexors are also outer-core mStanding exercises demand the most from your core musculature.
If you're seated or lying down, your body uses whichever surface you recline on to create stability and the core muscles gets lazy. To make the core stronger, you only need to let it do its job and protect the spine when you put heavy loads in your hands or on your shoulders.
From this perspective, every exercise is a core exercise. Complete an exercise with good form and you have trained the core to do its job. Consistently increase the load of an exercise using good form and you make the core stronger.