Electricity is notoriously difficult to store. The main ways we do it are with batteries; rare earth batteries are quite expensive and are commonly only used in small sizes like AAA, AA, D, and 9 volt batteries. There are several problems with these batteries; the extraction processes to create them are environmentally unfriendly, they have a finite lifespan, and they tend to slowly lose charge when they are sitting around. Lead-acid batteries like car batteries are also used; they are the common storage method for storing power produced by photovoltaic (solar) cells. They can handle larger charges and are stable for longer storage periods than rare earth batteries, but they tend to take up a lot more space and are quite heavy. The other important thing to note is that batteries can only store DC (direct current) electricity. Converting DC to AC (alternating current) for home use requires additional equipment, and some of the electricity is lost in the conversion process.
Pumped hydroelectricity is an indirect way to store electricity. In this technology falling water is used to generate electricity, and any excess electricity that is not needed right away in the grid is used to run pumps, which pump some of the water back up into a reservoir where it can be held until electric demand on the grid rises, at which time it is released and generates electricity again.
A new idea which is similar to pumped hydroelectricity is the concept of compressed air storage, where excess electricity is used to pressurize air; when the air pressure is released it can turn a turbine to produce electricity. This is still an experimental technology, and will required large volumes of space to be practical; some have suggested using natural underground spaces after the oil of gas has been pumped out of them.
The link below will take you to a study by the US Department of Energy on this topic.