A republic is a form of government in which power is delegated to a particular group by law. This is in contrast to those systems of government where the ultimate power resides with the head of state, typically a monarch, theocrat or dictator, rather than a set of laws. Republican forms of government are the antipode of hereditary monarchy.
In the case of democratic republics, power is usually held by elected representatives of the people whose powers are derived from (and limited by) a constitution. In other republics, this could be delegated to a traditional group of elders such as the jirga in the transitional government of Afghanistan. The leaders of Islamic republics such as Iran are typically a group of elder clerics who govern in accordance to Islamic law. The mostly extinct class of People's republics are socialist countries which pretend to be governed by the people but where in reality the people have very little power. In most cases (if not all), a non-democratic republic is essentially a form of aristocracy since all the power is in the hands of a group of elites. (And a lot of democratic republics can be seen as slowly going in the same direction due to the existence of an entrenched political class.)
 Republics and democracies
In the United States, a slogan popular among conservatives is that "the US is a republic, not a democracy," sometimes served with a wink towards the names of the two major parties (the Republican and the Democratic). The two labels are not mutually exclusive, though, as nothing prevents a republic from being a democracy. This false dichotomy is usually justified by using a very narrow definition of "democracy" - a pure direct democracy with no elected officials would not be a republic. What those people fail to realize is that nowadays "democracy" is usually a shorthand for "representative democracy with a constitution and/or separation of powers" (guess which was the first major one). It should also be noted that in modern times, there are only two types of functioning democracies - democratic republics and parliamentary monarchies (such as the UK). Republics don't prevent tyranny of the majority, constitutions do.
Republics are usually headed by a President. The powers of presidents vary wildly, from mere figureheads (for example, in Germany) to absolute dictator. Figurehead presidents are often appointed by the national parliaments, dictators often force their way to power, and the rest, with limited powers, are usually popularly elected. There are a few hereditary presidencies, but these are de facto inheritances rather than constitutional provisions. The UAE president, for example is customarily held by the ruler of Abu Dhabi—itself a hereditary position.