I've learned from Wikipedia that the British king or queen has the right to veto laws, but they don't use the right. Why is that? Is it because of tradition, respect for democracy, some regulations or perhaps something else? I'm asking because the presidents of my country (who are however elected by the people) have an analogous right, and they use the right quite often.
The monarch of the United Kingdom is, as are most contemporary monarchies, a historical artefact, with little (if any) political power, their role is largely ceremonial. Refusal of royal assent is rarely exercised any more, and the main reason is, as you suspected, respect for democracy. Conversely, the last time royal assent was refused in the UK was in 1708, when Queen Anne vetoed the Scottish Militia Bill.
Elected officials, on the other hand, are not going against democratic structures if they exercise their veto powers.
Because to do so would cause a constitutional crisis. As the Wikipedia article on the subject lists, previous such crises have more typically been caused by the head of state acting against the advice of the government on matters such as dissolving Parliament, or dismissing or appointing Prime Ministers.
However, the underlying issue is the same: in countries and states which have this system of government, where the head of state is expected to exercise what limited powers he or she has only on the advice of the government, when he or she doesn't, it can result in any of: fierce debate, governmental paralysis, fresh elections, a change in the constitution, the removal of the head of state (or the head of the head of state), or civil war.
In the UK and other constitutional monarchies, as well as in a number of Commonwealth countries and their constituent states, constitutional convention dictates that the Queen (or equivalent in other countries and states) is required to assent to bills presented to him or her after they have been agreed to by the legislature.
Although this power appears ceremonial, and in the vast majority of cases the Queen (or equivalent) does indeed grant assent when requested to do so, the power is real. This is demonstrated on the rare occasion when assent has been refused.
To give one such example relating to King Baudouin of Belgium:
In 1990, when a law ... liberalising Belgium's abortion laws, was approved by Parliament, he refused to give Royal Assent to the bill. This was unprecedented; although Baudoin was nominally Belgium's chief executive, Royal Assent has long been a formality (as is the case in most constitutional and popular monarchies). However, due to his religious convictions, Baudouin asked the Government to declare him temporarily unable to reign so that he could avoid signing the measure into law. The Government ... complied with his request on 4 April 1990. According to the provisions of the Belgian Constitution, in the event the King is temporarily unable to reign, the Government as a whole fulfils the role of Head of State. All members of the Government signed the bill, and the next day (5 April 1990) the Government declared that Baudouin was capable of reigning again.
UPDATE 24 Jan '13: The Queen and Price of Wales (the heir to the throne) also have to give their consent to bills which relate "to royal powers and the interests of the Crown and the Duchy of Cornwall", and this must be done during the bill's passage through Parliament.
As with Royal Assent, this is largely a formality. However, recently revealed documents show that the Queen did in fact refuse consent on a Private Members Bill in 1999, as well as on a few other instances prior to that.
While this may look like the Queen vetoing bills she didn't like, it appears that it was actually the government using a rarely-used power to prevent inconvenient bills from being discussed in Parliament.
Because there are more effective ways for Her Majesty to change or veto laws. The Queen can indeed veto a law after it has passed the Houses of Parliament, but it would be ill-advised. Instead, she can use her considerable "soft power" to warn the Prime Minister of her disagreement with the law before it is voted upon. Such warnings are secret, but are documented, and sometimes released. In the case of the Queen of Australia, concerning a move by a 1979 NSW Government to terminate appeals to the Privy Council, constitutional expert Anne Twomey remarks:
What is most remarkable about this story is not that the Queen, supported by the British government, was prepared to override the will of the New South Wales Parliament as late as 1979, but that this entire incident was hushed up so that the public never knew of it until at least 2006. It illustrates the immense “soft power” of the Queen. She does not have to refuse assent. It is enough to indicate that she might do so and in almost all circumstances a government will yield. This system is supported by the procedure for receiving advice from the realms. Informal advice must first be sent and it is only once the Queen’s private secretary indicates approval of the informal advice that the formal advice may be given. Hence, any objections by the Queen are raised and resolved at the informal advice stage so that the Queen never rejects formal advice.
The Monarch of the UK (and the Commonwealth Realms) is a ceremonial position and so she cannot, in reality, exercise this power. The only power I can imagine her exercising (beyond normal day-to-day duties such as awarding charters, awards etc.) is dismissal of government as that just means that another general election would be called. The problem being would be the vacuum for the interim government and the lack of time for campaigning.
Refusal of royal assent (vetoing laws) would cause massive uproar and would show little respect for democracy and therefore the people's choice. The likelihood is that the people of the United Kingdom would call for reform and to get rid of the monarchy but this would cause countless problems:
The monarchy generates massive revenue for the economy (through tourism etc.) and royal celebrations quite often bring in mass tourism and people of the realms (especially the British) spend money for souvenirs etc.
It would also cause a massive problem and would cost billions to dispose of the monarchy. Thousands of jobs would be lost, many organisations such as the Royal Mint, Royal Mail etc. would require name, structural and governance changes. The military governance & naming would change which would require renaming of almost all naval vessels and aircraft. C
City status balloting would need to be revised and the house of lords would fall into disarray as all awards and titles such as MBEs, OBEs, Knighthoods, baronships, earlships etc. are awarded by the Crown.
You also need to bear in mind that she is head of state for the Commonwealth realms also, not just the UK. So essentially the UK would be leaving the Commonwealth realms to dispose of her. This would create massive problems for other countries.
New systems would need to be established with regards to the appointing of governments, the final signing of bills. This would mean the many problems created by the dissolution of the Crown could not be rectified. The amount of problems caused and the amount of money that would need spending to fix it would be too significant to even contemplate.
To return to the original point, I doubt the monarchy would ever use a veto as this would cause dissolution of the monarchy which would end very very badly.
One simple way to perspectivize it:
The French Revolution was particularly violent, being driven by the outrage of the people against the monarchy's acts against the will of the nation (its people). Most other European monarchies were exposed to the same thing around the same time.
Everyone in the now European countries had become fed-up with their will not being properly represented.
Most of the EU countries allowed their royal families to stay, provided that they did not oppose to the democratic process and power. Some of the EU countries forbade their royal families from staying: Greece, Italy, etc.
The Queen and prince Charles have the soft power to stop progression of a bill in early stages before it goes to the house of commons and lords. the prime minister have private meetings with the queen weekly and no one is allowed to know what they discuss and spying of the PM leaking information is a very strongly punished crime. they also talk to the government and misters in private meetings. Also because the queen is not only the head of politics but also the head of religion because there is no separation of church and state, 26 bishops including the archbishop of Canterbury have a seat in the upper-house of parliament who are the last stage before an act of parliament reaches the stage where they need royal assent to pass it. they therefore influence laws at a more private and in early levels. they Royals have served the nation and those who know them realize they always act in good faith for example the prince of wales didn't stop the bill that forced him to pay higher taxes because of the land he owned and his property of Cornwall instead he agreed but that money was money he used for his Charity that has done immense good for the nation. also the prime minister can advise that the queen veto a bill which will be public and is a tool that protects the prime ministers power and will not cause controversy because she is acting on the advise of her ministers and the prime minister wanted to stop a bill that was passed against his will due to the opposition party and corruption in government.