Statics
A branch of engineering uses statics when they build bridges or buildings. Dynamic equilibrium is used as the civil engineers have to consider the forces that will be applied to these structures. By calculating the forces that are applied the engineers will maintain the balance of the structure against whatever force that is being exerted, and thus maintaining a state of equilibrium.
Draining of a bathtub
When a bathtub is filled to a certain level, and then the plug is taken out while water is still being added through the tap, the water that is being taken out in theory, will equal the water that is being put in. The amount of water in the tub will remain at a level that is constant, while there is a fluent amount of water being added to the bathtub system. However, if the system changes at all, such as the flow from either end of the tub, then the level of the water will also change depending on how much of change in the system occurred.
Salinity of the ocean
Salinity is maintained at a fairly stable level as dissolved solids are always being added through the streams, but are also being removed by many chemical, physical, or biological ways. The average time, otherwise known as the "reference time", is the time taken from input to output, and is different for each constituent, such as sodium ions or magnesium ions, this results in a different balance between the input and output rates of the different constituents.
Homeostasis in warm-blooded mammals
Dynamic equilibrium is incredibly important warm-blooded species, ourselves included. Homeostasis keeps our body temperature nearly constant. If we get too hot, then our body will cool us down by sweating as the blood vessels near the skin's surface dilate to allow heat to escape. If we get too cold we will shiver to create metabolic heat and the capillaries become smaller to retain heat and keep blood away from the arms and legs to maintain the core temperature.
Economics
Economically in a free market, the theory is that the price of a product remains the same, or equilibrium price, this is when buyers can buy all they want and sellers can choose what they want to sell. When the conditions of supply and demand change, so will the price by either decreasing or increasing.