A simple method for the ordinary beekeeper
Many "ordinary" beekeepers only have a few colonies and using natural queen cells for producing new queens may be adequate. There are others who may wish to progress to slightly more advanced methods where "artificial" queen cells will be needed and they will need to set up colonies to build them.
Some of the queenright cell building methods are a bit complicated and often need dedicated colonies or purpose made equipment, such as special boards. A queenless colony is an ideal way of producing a single batch of queen cells with several benefits and no extra equipment needed.
This is achieved by simply taking the queen away from a colony, when the bees will build queen cells under the emergency impulse. It is probably safest for the inexperienced beekeeper to wait 8-9 days, then remove emergency cells and add the prepared larvae. It may be beneficial to wait a couple of hours until the bees realise they are hopelessly queenless, otherwise they may reject some of your introduced larvae, but I have never found a need to.
It is possible to introduce larvae a couple of hours after the queen has been removed and at any time later, but there is a chance that emergency cells will be built on the existing larvae. These may be in advance of your introduced larvae and if one queen emerges from one of these cells she may destroy all your queen cells, or the colony may swarm with her. An experienced beekeeper may be able to manage this, but one tiny emergency cell that is hidden can cause great disappointment for the less experienced beekeeper. If prepared larvae are introduced to the cell raising colony soon after dequeening, I suggest inspecting the colony and removing emergency cells 4-7 days afterwards, by shaking all the combs - and again when the Q/Cs are distributed. Other emergency cells may be built, but they will not interfere with your introduced Q/Cs.
This method is good for the ordinary amateur as it works very well and doesn't take a colony out of honey production. A colony with a poor queen can be used and you can cull her when you remove her. If you use a colony headed by a good queen, then remove a 2 frame nuc and leave the queen in that as insurance in case something goes wrong, or requeen another colony with her. Either way you keep a potentially good queen.
The colony needs to be strong and prosperous, with the brood combs well covered with bees. You won't get good queens if you try to produce them under less than ideal conditions. A honey producing colony is ideal, but not essential. There needs to be adequate liquid stores, preferably unsealed, and pollen. It will be frowned upon by some, but I have often used a smaller colony, which is O.K., providing it is bursting with bees. A general rule if the colony is less than full size is to allow it to raise one Q/C per frame, so a 5 comb nuc should not be expected to build more than 5 Q/Cs. I repeat - it MUST be strong in bees. If there is no nectar coming in a gentle feed is beneficial.
A colony that is preparing to swarm is in an excellent condition to raise queen cells. Remove the queen and all queen cells and introduce larvae as above.
In my experience a queenless colony won't raise many Q/Cs if given a second batch, so when the first is finished, leave one Q/C and leave the colony alone until you expect the new queen to be laying.
This method will result in a brood break, which is ideal for varroa control. Get your timing right, but at some stage there will be a high number of mites on adult bees. An empty frame, preferably drone, that was put in the centre of another colony a week before will have a lot of unsealed brood. This can be put in your cell raising colony, then removed and put in the freezer when it is sealed.
A strong colony can be split into several nuclei and each given a Q/C.
On several occasions, especially if I want increase, I have made up a colony by removing a colony that is strong or preparing to swarm. In it's place make up a full colony with a couple of frames of food and frames of largely sealed brood covered in bees from several colonies. In a week or so they are bursting with young bees and are in a good state to raise Q/Cs. When you have finished they can either be left as a honey producing colony, or split and given a Q/C each.
There are many more possibilities when you have queen cells and a queenless colony.