Every object is made of particles (atoms and molecules), and those particles store energy. The energy stored inside a system by its particles is called internal energy. It is not the energy of the whole object moving, such as a ball flying through the air. It is the energy of the particles themselves.
The particles store energy in two ways. They are always moving, so they have kinetic energy. They also have potential energy, which depends on how the particles are arranged and the forces between them. Internal energy is the total kinetic energy and potential energy of all the particles that make up the system.
Heating a system increases the energy of its particles, so its internal energy goes up. This does one of two things. Either the temperature of the system rises, because the particles gain kinetic energy and move faster, or the system undergoes a change of state, such as ice melting, where the energy goes into changing the potential energy of the particles.