Supply Demand and Equilibrium
First of all, a clear distinction must be made between supply and actual sales, on the one hand, as well as between demand and actual purchases, on the other hand. Supply is the intention to sell a given quantity of a good, given its price. Demand is the desire to buy a given quantity of a good, given its price. A market is the event that buyers and sellers of a certain good meet together to trade. A good, in its most general definition, is everything that can be traded. Others prefer to distinguish the goods between material goods (apples, cars...), on the one hand, and services (health care assistance, music concerts...) on the other hand. Here, material goods and services are collectively referred to as goods.
For a given price, say one dollar, the demand (i.e. the amount of good the consumers want to buy at that price) may exceed the supply (i.e. the amount of good the producers want to sell at that price). If the exchange would occur at such a price, the consumer desires (plans) would not be wholly realized. On the other hand, the price of a good could be such that the supply of it exceeds the demand for it. In this case, the producers intentions (plans) could not be entirely realized. Actually, it is easy to understand that for many prices the supply and the demand of a good do not match, so that the plans of producers and consumers are inconsistent between them and cannot be realized. In fact, a market is said to be in equilibrium at a given price only if the demand for a good at that price equals its supply (at the same price). The price is therefore called 'equilibrium price'. 'General equilibrium' is the term used in economics to indicate that the markets of all goods are in equilibrium.
Microeconomics primarily focuses on individual choices: why, at a given price, a person demands a certain amount of a good? Why the producers (i.e. the people involved in the production of a good) are willing to sell, at that price, a (possibly different) quantity? What is the aggregate (=market) demand and supply of a good at a given price? What would be the situation at a different price? More importantly, what happens on the market when consumers and producers meet? Is there an equilibrium price for that good? If it exists, can it be reached? Are the many different markets linked in some way? Is there a general equilibrium, that is a set of prices such that all the markets are in equilibrium?
Even though the primary focus of microeconomics is on individual behaviour, its ultimate aim is to provide a simplified mathematical representation (=a model) of the working of a market economy. The scientific interest in the working of a market economy comes from our everyday experience: so many distant events are (or, at least, seem to be) coordinated amongst them to provide us what we need. How many people have worked to give us all the goods we consumed in the last two weeks? Adam Smith suggested that the market is the coordinating mechanism in the backstage, working as an 'invisible hand'.
Roughly speaking, a market economy is a state of the world such that the individuals may freely trade goods between them. For example, in a market economy a person can freely decide to sell her/his labour services to an employer but cannot be obliged to work against her/his will. Thus, the existence of a full-fledged market economy requires laws and institutions (for example, slavery is banned, theft is punished, property rights are protected etc...).
Along with the positive study about the actual functioning of a market economy, a crucial normative question is whether its outcome (be it a general equilibrium or not) is the best attainable result for the society as a whole.