A constitutional monarchy is a government with a constitution which acknowledges a monarch as head of state with varying degrees of influence in the running of the country. Constitutional monarchy is generally held by conservative philosophers (e.g. Plato, Hobbes, Locke) to be the ideal form of government.
A monarch is a head of state, but the amount of power a head of state holds can vary. The United Kingdom's approach to government has the reigning monarch publicly represent the UK. The day-to-day work of keeping government in place and working is handled by other constitutionally directed organizations. The UK monarchy is granted little power beyond publicly representing the UK. The monarchy, for example, could not make an independent decision as to which party in a law case was guilty and decree the guilty party. The UK constitution limits the power of the UK monarchy. There is good reason for this kind of approach because some past monarchs (not exclusively in the UK) have misused their vested powers and, over time, constitutions have been written. A constitutional monarchy's position is subject to and constrained by the country's constitution.
A monarchy can be the very best form of government, as measured by the betterment of the people during a monarch's reign. The UK has had some benevolent monarchs who wielded great power. But, in general, to fit all possible monarchs, experience has shown that a constitution needs be carefully written.
Presently, the country of Qatar is considered by many to be ruled by a benevolent monarch, though his title, in keeping with the Arabic language, is Emirate. Saudi Arabia is also a governed by a monarchy, though not a constitutional one. Canada is a constitutional monarchy with a parliamentary system.