Generally speaking, a spontaneous process is one that occurs without the influence of external forces. A common example that is used to portray the difference between a spontaneous and nonspontaneous processes is the dropping of a breakable object like a beaker. The beaker will fall and break unless outside forces are used to stop it. However, the opposite process, a broken beaker being reformed into its original condition and defying gravity to lift into the air, is nonspontaneous. The beaker cannot simply reform and lift into the air on its own, it requires a skilled glassworker to fix a broken beaker and someone to lift it in the air.
Chemically, when we use the term spontaneous, we are referring to any change that moves a system toward equilibrium. The term spontaneous does not imply anything about the speed or rate; this is the domain of kinetics. Spontaneity simply provides information on the direction of a reaction.