Drum Printers are some of the fastest printers made before the advent of modern super-high speed laser printers. Models in the early 1990s could run at speeds of over 6000 lines per minute (yes, 100 lines per second or about 90 pages per minute). By comparison, the fastest dot matrix printers were pushing towards 200 characters (2-3 lines) per second.
The principle behind a drum printer was quite simple. The main print "line" had a row of hammers, one for every character position on the paper (which would often be 132 or more for wide carriage paper). These hammers would be in front of the paper, and between the hammer and the paper would be the ribbon. Behind the paper was the drum. And on this drum would be impressions of fully formed letters and numbers. There would be a whole row of A's, then a whole row of B's, and so on, usually for upper case and lower case letters, numbers, and some punctuation. The paper would be pulled through by sprocket feed, usually from above and below.
(The arrangement could be reversed, with the hammers behind the paper and the drum in front - if this was the case, all the letters and numbers of the drum would be mirror images.)
For every line that needed to be printed, the drum would do one complete revolution. If, say, there had to be a "J" printed in the fifth position, then when the drum was on "J", the fifth hammer would fire, imprinting a "J" on the paper. By the time the drum had revolved, every possible character had been covered, so the line would be complete. The paper would then be advanced, and the process started again. With the drum spinning at 6000 rpm, and a fast paper advance, speeds of close to 6000 lines per minute could be achieved.
The output was limited to the characters on the drum (like a daisy wheel printer), and the print quality was far from brilliant. And the noise. Let's just say that most drum printers were kept in soundproof boxes! But as I mention above, the speed was outstanding. So if you were in a company that had lots of large reports, or payslips, or anything of the sort, to print out, they were the best option.
rootbeer277 correctly mentions that like dot matrix and daisy wheel printers, you can use carbon paper to make multiple copies on a drum printer, as it's an impact printer, whereas you can't do this with inkjet or laser printers. However, you'd probably have to use special carbon paper to cope with the high speed motion of the system.