Mauritius is the most reliable producer of fruit. Some say that Mauritius has a very slight cinnamon-like taste. If you only have enough room for 1 tree and you really want a crop every year, Mauritius is probably the best variety for you. This variety is rather small, both the fruit and tree size.
I've read conflicting opinions about the taste of the Mauritius variety. Apparently the flavor has a noticeable difference from the other lychee varieties, kind of slightly sub-acid taste. Some people have said the flavor is very good to excellent, even that it is their favorite lychee variety they have tasted. Other people have said it is the only lychee variety they don't like.
Emperor has the largest fruit size, up to 1½ to 1¾ inch diameter, but its flavor is a little bland/insipid compared to other lychee varieties.
No Mai Tsze is supposedly the best tasting, but has small fruit size and is not the most reliable at producing fruit every year.
Brewster is an all-around good variety, but not really exceptional in any single category, the tree of this variety can grow quite tall so it may not be as suitable for small yards. This is the most common commercial variety in America, and I think the fruit exterior of Brewster has a more attractive appearance than the other varieties, smooth and bright red.
Kaimana is supposed to be an all-around good variety, with great taste and large fruit size, combined with a 'chicken tongue' seed so the fruit contains more edible meat. From what I have seen, Kaimana can get pretty large fruit sizes also, and may possibly contain more edible meat than the Emperor variety (it has a smaller seed). This variety would be a lot more popular if its fruit production were more reliable. From what I have read, there will be many years where it will not produce.
Some lychee connoisseurs are real fans of Hak Ip, but others think the taste of this variety is too "medicinal". It is prized for the firm texture of its flesh and the lack of juiciness.
Mauritius, Emperor, and Hak Ip are "mountain-type" varieties, so the trees do not grow as big and may have slightly more cold and drought tolerance than other lychee varieties (though all lychees need consistent water in dry climates).