Step 1 - Understanding what "reversible" means
Suppose we have a sample of water. If we warm it up, the water will become vapour. That's what we call a physical process.
But this is not a unidirectional physical process. In much the same way, we could cool down a sample of water vapour, thus obtaining liquid water once again. This would be a reversible process.
In nature, both things are happening at the same time. In the surface of a lake, for example, water is becoming vapour at the same time that water vapour in the atmosphere is turning to liquid water. This is a situation we call physical equilibrium.
Step 2 - Can we have chemical equilibrium as well?
In a chemical reaction, chemical bonds are broken in order to form new chemical bonds. The atoms are rearranged thus forming new molecules.
At the same time, the newly formed chemical bonds in the products can also be broken, and a new atom rearrangement would occur, restituting the reactants.
A chemical reaction, therefore, can be reversible and, in fact, most of them are reversible. Equilibrium is reached when the direct reaction and the inverse reaction are happening with the same velocity.