Kinetic energy is the energy due to the motion of the body. If the body has a mass (m) and a velocity (v), its kinetic energy can be calculated as 1/2 mv^2. Potential energy is the energy by virtue of position of an object. If the mass of the object is m and it is at a height of h (above a certain datum, typically ground), its potential energy ca be calculated as mgh.
It is possible for an object to have both these energies simultaneously. The condition for that is the object should be in motion and should be above ground. An example is a stone falling from a height. It is in motion, hence has kinetic energy and it has potential energy, since it is above the ground. Another example is a roller coaster. It is in motion, hence has kinetic energy and it goes up and down and hence possesses potential energy (as it goes up, it increases and it decreases as the roller coaster comes down). One can also think of a ball thrown upwards as a relevant example.
Hope this helps.