A drop handlebar is one in which the middle of the bar is the highest point, or nearly. Most bicycles built for fast or long-distance riding have drop handlebars, which provide a range of different grip positions, allowing the rider to change positions for variety and to accommodate different road/wind conditions.
The most common style of drop handlebar is the "Mæs" bend. Variations include the "randonneur" and "anatomic" bends.
The main advantage of drop handlebars is that they offer several different hand positions. For longer rides, the ability to change positions is very desirable. Riding for a long time in any one position tends to be uncomfortable.
People who think they don't like drop handlebars are often actually objecting to the position of the bars on the drop-bar bikes they have tried.
Bikes with drop bars often have the bars mounted rather low and far forward, so that the rider has to lean forward quite a lot to reach the bars, especially the lower "drop " position. If these people tried a bike where the drop bars were placed higher, and closer to the saddle, they might find they really liked them.
I have an article on this topic, called "Hands Up."
Most newer drop handlebars come in either single-groove or double-groove versions, with grooves along the upper section to acommodate brake (and shift) cables running under the handlebar tape.
These grooves are totally unnecessary. Indeed, the double-groove variety is often rather uncomfortable due to the rear groove's causing sharp ridges against the rider's hands.