What is it like to live under a monarchy?

It all depends on what kind of monarchy one lives under. If you live under an absolute monarchy, for instance, you are a subject, but one totally without rights. You only enjoy, if that's the right word, privileges granted to you by the king or queen. As the monarch is absolute he or she exercises absolute power. As such, you have no say whatsoever in the government or the future direction of your country. Laws are little more than royal edicts, made purely and solely in the interests of the king or queen. If you don't like these laws, then tough; you have no right to complain or protest. And if you dare to try, then you'll be punished for treason or sedition. As the monarch is absolute, such punishment can take many different forms, some of them positively blood-curdling. Torture can also be used to extract confessions from those suspected of crimes against the state. An absolute monarchy is a rule of men and women, not of laws.
A constitutional monarchy is a different ball game entirely. The monarch has a role to play but it's usually no more than formal; the king or queen is essentially a figurehead, representing a sense of historical continuity. The idea is that a notionally impartial figure at the head of state provides an element of stability amidst the turmoil of ever-changing governments.
The monarch formally approves laws, but plays no part in drafting them. In the United Kingdom, for example, laws passed by Parliament are given what's called the Royal Assent before they reach the statute book. The government is formally known as "Her Majesty's Government," but again, this is purely formal. In substance, the government is conventionally formed from the largest party in the democratically-elected House of Commons. The leader of this party must formally obtain permission from the Queen to form a government, but the request is never denied. The Queen is there to provide a measure of continuity after the cut-and-thrust of a contentious General Election campaign.
On the whole, life under a constitutional monarchy is infinitely more agreeable than that under the absolute variety. For one thing, you have a say in government. And that government operates under the rule of law, not the mere whim of a monarch and his or her advisers. Under a constitutional monarchy you're still technically a subject, rather than a citizen, but you do have formal legal rights, though they differ depending on which particular country you're talking about. In any case, you normally have the perfect right to criticise the institution of monarchy itself and can actively argue and campaign for its abolition. You can also voice the opinion that it might be more modern, more democratic, and considerably less costly to have an elected President instead of a king or queen as head of state.