Several methods to raise queens
This page links to methods by which queen honey bees can be raised in order to replace ageing queens, those whose colonies are of undesirable characteristics or to head additional colonies. Such methods are described as "queen rearing" or "queen raising". This should not be confused with "bee breeding" which is the process of selection that is used to decide which colonies queens should be raised from and which queens to cull.
Many beekeepers think that raising queens is just a matter of splitting an existing colony and letting the bees do the rest - what the Americans call a "walk away split". At a very basic level this may be true, but if we seriously want to improve our bees, we need to put in a little more effort and rear queens from colonies that exhibit desirable traits.
Having said that I do not wish to put off any "would be" queen rearers. Queen rearing is not difficult and it is not a mysterious "black art", despite the complicated way some lecturers and writers describe it. Rearing good quality queens that you have bred yourself is a satisfying part of beekeeping that I recommend to all. The methods you use need not be complicated, as the bees often provide us with natural queen cells. We just need to decide which colonies we want to use them from and which to avoid.
There are many more methods of raising queens than are listed here. None are hard to master, though many modern people, whether they are beekeepers or not, seem to struggle with practical skills. An example is grafting, where in my experience of running workshops, only about 20% manage it with any degree of success. For some reason the other 80% don't seem to possess the required amount of dexterity.
There is a lot of information about queen rearing, bee breeding and bee improvement on this website, with regular additions. Please spend a bit of time exploring them. There is much more on the pages accessed by the buttons on the left.