Evaporation happens when some molecules present at the surface of the liquid gain sufficient kinetic energy to overcome the binding inter-molecular forces of its surrounding molecules and break away to form a vapour. This raise in kinetic energy can take place due to a slight rise in the temperature of the surroundings or even by molecules colliding with each other. Evaporation can take place at any temperature and it is a very slow process.
On the other hand, boiling is defined as a fixed temperature at which a liquid changes its physical state gas. It is actually a point where the vapour pressure of the liquid same to the atmospheric pressure.
Molecules of a liquid continuously keep colliding with each other and as a result, their kinetic energy increases. When a certain molecule has such increased kinetic energy and it reaches the surface of the liquid, it overcomes the attractive binding force of other molecules and evaporated into the atmosphere. However, in case of boiling, all the molecules start gaining kinetic energy due to the heat provided to the liquid and try to force their way out into the atmosphere. This boiling process happens throughout the liquid, not just at the surface.
Evaporation can take place at a temperature well below the boiling point of a liquid, as only few molecules can gain sufficient kinetic energy to break away. Boiling occurs when the average kinetic energy of all the molecules reach a fixed temperature and it is only then they have sufficient energy to convert into vapours.
Evaporation is a very slow process because rarely any molecule gains enough kinetic energy through collisions to break away from the surface. On the other hand, once the liquid reaches its boiling point, the evaporation process occurs at rapid pace, as all the molecules posses increased kinetic energy.