How do we know whether a chemical reaction is reversible or not?

There are a couple of ways that a reaction becomes irreversible. Generally, when a product of the reaction is eliminated from the system, you can consider it to be irreversible. How can a product be eliminated from the system? Normally by altering its physical state: Precipitation of a product may be considered an elimination if the equilibrium concentration of the saturated solution is very low. Similar for generation of gases in a solution. Another strong clue for irreversibility is the decomposition of large, complex compounds into lots of small, very simple products. Chances that the complex compound will generate from mixing those little products usually are low (e.g. hydrolysis of hexa-methylene-tetramin, or combustion of candle wax). From a physical chemistry viewpoint, reactions tend to be irreversible when the reaction happens without equilibrium.

All chemical reactions are reversable. What most people want to know is how easy it is to reverse the reaction. This depends on the difference in chemical energy between the reactants and the products. If the difference is very small, then the reaction is easily reversable. Chemical energy is mesaured in a calorimeter. It burns substances and the change in heat is carefully measured. The chemical energy is then calculated from the amount of heat generated.

if it is reversible, then you will end up with the same mixture (including both products and reactants) whether you start with 100% products or 100% reactants. But are you asking how you know if a process is reversible under particular conditions? In that case, it is reversible if and only if it is carried out in such a way that equilibrium is maintained. For example, melting ice, if you supply heat fast enough to melt the ice but without at all raising the temperature of the water. You can probably see that "reversible conditions" is an ideal situation, never achieved in practice.

it is reversible if the Gibb's free energy, or Delta G is 0 at a specific temperature. If delta G is negative, then the reaction is spontaneous at that temperature. If it is positive, then the reaction is non-spontaneous at that temperature