How is it that you can have both water and ice at 0 °C and both water and steam at 100 °C?
This requires understanding the difference between internal energy and thermal energy . According to the kinetic theory of matter, matter is made of very large number of atoms/molecules which move, vibrate and rotate randomly. There are kinetic energies associated with these random motions and potential energies associated with the atomic/molecular interactions.
Internal Energy (U) : The sum total of kinetic energy of random motions of all the particles and the potential energies associated with all interacting pairs of atoms/molecules in a system is called its internal energy.
Thermal Energy : By thermal energy we mean only the kinetic energy part of internal energy. Thus thermal energy is a subset of internal energy. It is the thermal energy which is related to temperature. Temperature is related to the average of the kinetic energy of random motions.
Ice - Water Transition : Ice is water in solid state, in which the molecules are strongly bound together by a combination of hydrogen and covalent bonds. There is potential energy associated with these bonds. In this state the water molecules can vibrate and there are kinetic energies associated with these vibrations.
At temperatures below the ice-water phase transition any external energy given to the system is used to increase the kinetic energy of the molecules and so the temperature of ice increase. In this case the change in the internal energy happens through the increase in the thermal energy.
When its temperature reaches the phase transition temperature all the energy put into the system is used to break the bonds and so its potential energy changes with no change in the kinetic energies. Since temperature is related only to the kinetic energies, there won't be any change in thermal energy though the internal energy itself increases. Liquid water is those molecules with the bonds broken. The average kinetic energy associated with liquid molecules is the same as the average kinetic energy of solid water molecules. That is how water and ice can be at the same temperature.
Similar explanation can be given to the Water-Steam transition. In the liquid phase water molecules still have a weak bond and in the gaseous phase more modes of kinetic energy , other than vibration, are possible (translational motion, rotational motion etc.)