Equity is "fairness" or "justness" and, specifically, is the name given to the whole area of the law that deals with disputes between persons when neither of them has done anything against the law, but there is a conflict between their rights or claims. Thus, it is to be contrasted with "law," which is the legal principles from the common law, the laws enacted by governments, and the "case law" (the principles set forth in courts' opinions deciding cases).
The "law courts" or "courts of law" were the courts all over England that enforced the king's laws in medieval times, but the "chancery courts" or "courts of equity" evolved from his discretionary decisions based on the specific facts of a single case. The only remedy a court of law can award is money damages, but a plaintiff whose neighbor will not return his only milk cow, which wandered onto the neighbor's property, for example, may want that particular cow back and not just its monetary value: A court of equity can order the neighbor to return the cow.
In the U.S. today, the federal courts and most state courts have combined both functions in the same courts, so a plaintiff can get legal and equitable relief in one proceeding. This reflects the position in England where the fusion of law and equity was substantially effected by the Judicature Acts 1873-1875. Delaware, however, still has separate courts for law and equity -- and its Court of Chancery is where most cases involving Delaware corporations are decided, so it shapes corporation law for the whole country -- and some other states have separate divisions for legal and equitable matters in a single court. It is important to state that in the United States where trial by jury is guaranteed it is not guaranteed in court of equity as equity can only be dispensed by a judge as it is a matter of law[?] and not subject to the intervention of the jury as trier of fact[?].
Besides corporation law, which developed out of the law of trusts, areas within the jurisdiction of chancery courts included wills and probate, adoptions and guardianships, and marriage and divorce. The kinds of relief chancery courts could grant included injunctions (which are court orders not to do something specific, like bulldoze a fence someone else claims is theirs) and court orders requiring a person to take some specified action (like sign over the deed for a certain parcel of real estate).
Whereas the laws a court of law enforced were often written down by whoever made the laws, the principles a court of equity applied were not, so a list of the principles evolved as the Maxims of equity. (Similar "maxims of the law" also exist.) Indeed, one of the historic criticisms of equity as it developed was that it had no fixed rules of its own and each Lord Chancellor (who traditionally administered the courts of equity on behalf of the King) gave judgement according to his own conscience. John Selden, an eminent seventeenth century jurist, declared, "Equity varies with the length of the Chancellor's foot".