You have several options:
- integrated shifter/brake units (brifters) which are modern road bike style.
- bar end shifters. These are not brakes
- Downtube shifters - these would be common on bikes from the 70s and early 80s.
- Stem shifters - a relocated downtube shifter, mid 80s to early 90s. Image is a single, but duals were common.
- rotary - this one I've never seen. It appears to be a MTB style "web-of-thumb" rotary control on a bracket mounted to the stem. Could be unique to Rolhoff IGH.
Another option is to not go to drop bars, but to shorten your existing flat bars. Its cheaper, but you won't get the variety of hand positions that a drop bar will give. Its also impossible to revert, once you've cut some off there's no going back. However you can use the existing controls, just shuffled inward a bit.
Any gear lever system you get will be either indexed or friction. If its indexed, it has to support the number of cogs on that end of the bike, probably 3 for the front chainring / left hand shifter, and 6 for the rear / right hand shifter.
If you chose a friction shifter, then its up to the rider to get the chain into the right place, and then friction holds the lever. Bar end shifters are often friction based. Brifters are always indexed.
Please do let us know what you ended up doing.
EDIT OJS makes an excellent point - road brake levers for caliper brakes than V-brake or a centerpull brake or a cantilevered brake; they will require more cable pulled than a caliper brake. So you may be fitting rim caliper brakes to your conversion as well. Does the frame have mounting holes straight over the tyre?
If you have hydraulic disk brakes, then this is getting really expensive quite quickly. Cable-operated disk brakes may or may not work with road brake levers... the cable pulled to activate them must be the same as the flat-bar brake levers.
So my suggestion is - don't. Look for a road bike that is like your requirements, and buy it. The red one in the first image is my bike, which cost $123 NZ. So I have a MTB still for towing the trailer or for wet weather or for biking on anything a little rough. Remember, N+1 is the correct number of bicycles to own.