The vast majority of people are on fixed electricity meters, which simply measure how much electricity has been used. These are used by the utility company to calculate how much you owe them for the electricity you have consumed.
During the day there are peak times when many people are using a lot of electricity - for example, in the early evening when many lights, tv's, and computers are switched on, and ovens and kettles are being used to prepare meals. In the middle of the night demands drops right off.
A smart meter offers two way communication between the utility company (electricity supplier) and the consumer. When demand is high (and therefore wholesale electricity prices are very expensive) the utility company can pass high costs onto the consumer. When demand is low (and therefore wholesale electricity prices are very cheap) the utility company can pass savings to the consumer. This motivates consumers to use electricity when demand is least so that they can save money.
This video shows the 'In Home Energy Display' by General Electric