Many Types can fulfil this role. Some use standard size brood frames, some use reduced size frames and others are specially made for the purpose.
The 5 B.S. frame nuc is common in the U.K. and can be used for many purposes, but it can be wasteful on bees if used only for mating purposes, especially in a larger apiary. A nucleus is a handy addition to any apiary and if used for rearing the odd queen it will provide a useful function.
The 3 frame National Nuc is 150 mm wide and was once part of the British Standard 1300 for beehives. Three of them side by side have the same size and shape as a standard brood box, so can be put under a national roof - not a great benefit in my experience. A useful way of mating queens in a smallish apiary, especially if they can be built up for increase.
A 2 B.S. frame nuc if made up early in the season is a very good way of mating queens. It is economical on bees and can have several uses during the summer.
Boxes that take half width B.S frames and bifold frames. are quite versatile but will probably have to be specially made.
Boxes that take one third width B.S. frames and triplefold frames were designed and made by Dave Cushman, but after using them for some time he thought that the half width option is better.
The Apidea, Kirchhain, Swi Bine and Mini BiVo are made from expanded polystyrene and require only a very small quantity of bees. The Kirchain is also known as Keiler, Segeberger and Warnholz, although I have no idea why. From time to time there are other polystyrene mating nucs that appear on the market, often to quickly disappear.
They are viable mating nucs, but in my view they need quite a bit of management to avoid failures. In the hands of experienced beekeepers who want reasonably large numbers of queens I think they are fine, but for the beekeeper who only wants the odd queen every so often, I think they may be disappointed.
Dave Cushman was a serious bee breeder who desired more space than these types provide for assessment of brood after mating. Albert Knight of BIBBA has come to the same conclusions and he has developed an adaptor board that will allow two Apidea hives to be used as a two storey single hive.
The Mino BiVo is a rather small unit and may be difficult to operate, but the page linked above details some adaptors and extras that make it more easy to use.
The Kirchhain nuc has been used a great deal in U.K. and Germany and the Kirchhain Adaptor Frames page details how the frames can be inserted into full sized hives, to get the comb drawn, or to have them filled with stores or brood.