Efficiency and equity are conditions of a mixed economy that are seen as microeconomic goals. These conditions are desired by society and the government tries to attain these goals by implementing economic policies. A certain market is considered efficient when there is no deadweight loss. There are two types of efficiency that economics is often concerned about: productive efficiency and allocative efficiency. Note that a market that is allocatively efficient is necessarily productively efficient, but the reverse is not true.
Productive efficiency occurs when a given market or economy uses the available resources to produce the maximum possible output. Allocative efficiency further includes the condition that the output produced provides the highest level of utility or economic surplus possible. One can imagine an extreme example of an economy that has the option to produce either garbage or diamonds. If the economy uses all its manpower and machines to produce as much garbage as possible, it could be considered productively efficient, but since no one is really getting any economic surplus from it, it is obviously not allocatively efficient.
It is important to understand that equity is not important when considering efficiency. Consider the two following market structures. First, there is a market for concert tickets and the ticket prices sell at equilibrium; consumers gain a certain amount of consumer surplus and the producers gain a certain amount of producer surplus. Second, the same market for concert tickets exists, but the producers manage to discriminate consumers through scalping without costing them anything. Producers manage to sell the tickets at the highest willingness-to-pay for each consumer and thus consumers receive zero conusmer surplus and all of the economic surplus becomes producer surplus. Obviously the first market structrure is much more equitable between consumers and producers, but both markets are considered equally efficient (the economic surplus is the same).