Marketing activities are those which are made to have financial gain in the form of earning profit /remuneration by selling the product produced by the activities of an economic agent.
Non-market activities are those activities which primarily aimed at self satisfaction or consumption.
Suppose, a cook cooking food for a restaurant is an market activity but housewife preparing same food in a house is a non-market activity.
Another example of market activity is teaching in an institution against a remuneration is a market activity but teaching your son/daughter in house is a non-market activity.