Simply put, thermal mass is the ability of a material to resist temperature change. This should not be confused with insulation materials which resist thermal conductivity.
Thermal mass is related to the specific heat capacity of the material which defines the energy needed to raise said material by 1 degree, per unit of mass. For example concrete has a specific heat capacity of around 880 J/kg°C, water is 4182 J/kg°C and air is 1005 J/kg°C. This may seem confusing at first because air has a higher specific heat capacity than concrete but it is easier to heat up. The key here is that specific heat capacity is a function of mass, in this case kilograms, and concrete is much denser than air so in terms of physical size 1kg of concrete is tiny compared to 1kg of air. It is easier to use volumetric heat capacity which takes into account material density by multiplying it by specific heat capacity.
This is useful for example in a hot climate where a material with a high volumetric heat capacity (it takes a lot of energy to make it hot), will stay cool and lower the operative temperature (the temperature that it feels) despite the air still being hot. For example if you were in a stone castle where the air is 80 degrees inside and outside, its going to feel much cooler inside the castle despite the air temperature being very similar.
Heat that is stored in thermal mass is heat that would be increasing air temperature if it wasn’t absorbed. Heat is transferred to thermal mass through conductive heat loss from contacting a warmer object, convective heat loss like air or water, or most commonly radiative heat loss where for example the energy radiated by humans is absorbed over hours. Then at night you can perform “Night Flush Cooling“ which means opening carefully placed windows or doors so that the heat store in the material can be flushed out at nigh. This is caused by the thermal mass transferring its heat to the cold air flowing over it through convection. In scenarios where night flush cooling isn’t enough to fully remove the heat energy built during the day, you may need to use air conditioning to aid the process.
In a cold climate thermal mass can be used to passively heat a space by placing where it gets lots of direct sunlight which will be absorbed into the material and release throughout the night. Then if a hot spell comes you simply cover the openings that were letting light onto your thermal mass, place some rugs and cover the mass to resize it properly and you have a passive cooling strategy instead!
A few important notes are that the sizing of thermal mass is very important. Too little thermal mass means the material temperature will be closer to air temperature and remove any benefits whereas too much thermal mass means the material will never gain enough energy in a day cycle to flush it out at night.