Starting with the word: dräll is a difficult word, probably coming from latin trilix, meaning three threads. (Note: in the etymology there is nothing at all that suggests "pattern".)
In Sweden, (the unqualified word) dräll has often been used as if meaning patterned (any kind of pattern). Sometimes, the word is qualified to "äkta dräll" (true dräll), when referring to a pattern where blocks of warp-faced structure contrasts with blocks of weft-faced (either turned twill or turned satin - thus multishaft).
As multishaft looms were not the most common, but the wish for patterning was, several methods for patterning on only 4 shafts were developed - as they were developed to mimic "true dräll" they often are referred to as "simplified drälls". Examples of simplified drälls are daldräll (overshot), jämtlandsdräll, sålldräll (Ms and Os, I think), kuvikas (S&W)... All of these have one principle in common: there is one warp system, but two weft systems, one of which gives plain weave.
There are many patterns named after a province (daldräll, jämtlandsdräll), a village (rovadräll, tynderödräll).
Other "meaningless" names are fattigmansdräll (poor (wo)man's dräll), soldatdräll (soldier's dräll). Another thing they all have in common is that the names do not really denote a specific technique...
There is also halv-dräll, which only has two blocks. There have been several "inventors" of that technique, one of whom was Wilhelmina Bergman. She published "Den lilla vävboken" about 1878, where she claims to be the "inventor" of a way to weave dräll on just 4 shafts (and she got a patent, valid for a number of years, 5, I think). This method is very much alike to what we call halvdräll today, with the difference that she did not use two different yarns for weft. (In fact, several old books do not) Over the years (the book was reprinted many, many times) she "invented" 4 ways to thread her... weave (she doesn't give it a name). Here is a screenshot of all four:
At some point another "name" cropped up: five-shaft dräll (femskaftsdräll). These were obviously done on five shafts (with a special five-shaft pulley, pictured here - scroll down a bit), and were woven in some kind of lace weave. As I am notoriously hopeless with the "names" of lace weaves, I'll leave it there... (An example can be seen on my website, here - at the end of the article.)
There were also a couple of ways to add shafts, dräll pulleys or "two-tier pulleys" (lunor, in Swedish). I have written about both kinds here. These were (still are?) quite common, even though countermarche looms had begun to be used already in the 1840-ies.
EDITED TO ADD: there is now a special page on my website which deals with dräll pulleys: http://bergdalaspinnhus.com/artiklar/cb-looms2-e.html