Many people believe that the term spontaneous in 'spontaneous reaction' is associated with the speed of the reaction in question, which is absolutely wrong. In fact, some of these reactions are so slow that they go on millions of years. The decaying of a diamond to graphite is an apt example of the same. Basically, the rate of reaction is not dependent on its spontaneity, but is dependent on its chemical kinetics.
What is a Spontaneous Reaction?
Spontaneous reaction is defined as a reaction which takes place on its own, without the involvement of any external factor to facilitate it. It is also defined as a time-evolution of a system in which it releases free energy and moves to a lower, but thermodynamically more stable state of energy. In rare cases though, an external trigger may be required at the beginning of the process to start it. But once it starts, it continues on its own. This is in stark contrast to non-spontaneous reaction, wherein some external factor has to continuously come into play to make sure that the process continues.
The direction of a spontaneous process is determined by the laws of thermodynamics, which govern the relations between states of energy in a closed system. Even though spontaneous and non-spontaneous, both reactions are possible, only the former occurs naturally, while the latter has to be triggered and supported. More importantly, the reaction in the case of spontaneous reaction is always of a non-spontaneous nature.
In chemistry, the concept is quite popular in context of endothermic and exothermic reactions. Most of the reactions that we observe are exothermic reactions, which are chemical reactions accompanied by the evolution of heat. That being said, there do exist some endothermic reactions, i.e., chemical reactions accompanied by the absorption of heat, which are spontaneous by nature.
One of the best examples is rusting of an iron nail. If you observe closely, you will realize that although the iron nail rusts slowly, it rusts continuously without any external factor coming into play. On the other hand, an example of reaction which requires a trigger is burning of wood. Once the wood is set on fire (trigger), it continues to burn on its own without any other factor coming into play so as to facilitate the burning process.
An example of quick spontaneous reaction is combustion of hydrogen, while an example of slow reaction is graphite turning into a diamond. While endothermic reactions absorb heat from the surroundings, exothermic reactions release heat into the surroundings. One of the best examples of endothermic spontaneous reaction is ammonium nitrate dissolving in water, wherein it absorbs heat and makes water/solution cold. On the other hand, one of the best examples of exothermic reaction is the chemical reaction between sodium and chlorine, which emits heat in form of flames.
To sum it up, this is a process which can takes place on its own under the given set of conditions. However, one needs to understand that the spontaneity in the case of such reactions refers to the feasibility of the process, and not its speed.