A system with constituents that are able to produce their own energy. For instance, they are often self-propelling. See also the wiki article. Due to the energy consumption, these systems are intrinsically out of thermal equilibrium.
If the system is made of bulk matter, it's called Active matter.
Examples of active systems are schools of fish, flocks of birds, bacteria, artificial self-propelled particles, and self-organising bio-polymers such as microtubules and actin, both of which are part of the cytoskeleton of living cells.
Dry active systems
A prominent example of active system are Active colloids
Biophysics (biological systems are active systems).
Flocks, herds, and schools: A quantitative theory of flocking. See Complex systems
DNA nanomachines in DNA nanotechnology