Most of the answers are correct, but I want to dispute some relatively minor points made some answers and comments.
Riding on the Hoods
Writers here are correct that most people are on the hoods much of the time, perhaps even most of the time.
@Mac and @juhist are technically correct that the hoods don't offer as much leverage on the brake levers as the drops position. However, if you have decent quality brakes (upgrade the pads if you have cheaper rim brakes, e.g. to Kool Stop pads), and the bike is reasonably well maintained, you should be able to conduct an emergency stop from the hoods without trouble. Maintenance comes in because after a while, contamination seeps into your brake and derailer cable housings, and this adds friction. It's worth changing your cables regularly. This comment isn't restricted to hydraulic disc brakes; I find that I can crash stop from the hoods on my rim brakes.
Other positions on the tops of the bars
The picture below is from a blog post by Velo Orange.
Basically, both the ramps and the tops of the bars are useful alternative positions when not descending or trying to ride very fast. Other posters have called the ramps the corners of the bar. I believe that ramps is the proper term, albeit it's not used often.
Some people have objected that many bars have the ramps sloped downwards to an uncomfortable degree. However, some bars are designed so that the ramps are a lot closer to level with the ground. For example, Rivendell is a purveyor of comfort-oriented drop bar and other bikes. They are advocates for the Nitto Noodle bar for precisely that reason: by default, the ramp is sloped down by about 15 degrees, versus what they say is 24-32 degrees for performance drop bars. Additionally, Rivendell advocate rotating the bar upward so that the ramp is sloped at -5 degrees. Rene Herse cycles (purveyor of, among other things, randonneur parts and bikes) makes a similar point about the Maes Parallel.
This blog notes that flat ramps may have originated in the 1950s. The angle of the ramps is a matter of preference; I actually prefer more downwards slope than the Noodle.
One issue is that there is no standard term that I know for the intended angle of the ramps (relative to the ground). Thus, it is hard to search for bars which are designed for the ramps to be flat when mounted. Zipp may call it the ramp angle, but I don't see this term used elsewhere. I am uncertain how modern, performance-oriented bars stack up. An earlier version said that I thought most modern drop bars were not designed for flat ramps. This is probably mistaken! The picture below shows three different handlebars offered by Zipp. The SL-80 is designed to have the flattest ramp when mounted; the angle figure behind the stem is the ramp angle.
Readers who do angle their bars up to reduce the ramp's slope should note that this will change the way the drops are angled, possibly making them uncomfortable to use.
The Case for the Drops
Generally, writers correctly said that the drops are for when you want to ride hard or descend. One writer said that the drops are less stable. I must disagree; your center of gravity is lower, and this should increase your stability. This is why people prefer to descend on the drops.
Riders may be less comfortable in the drops, or they may perceive them as less comfortable. While nobody may want to do a full century ride in the drops, I think that most riders with physical function and flexibility in the normal range should be able to achieve a comfortable drops position. One should be able to alter the fit and change components to achieve a comfortable bar position. Heine feels that in general, it is a myth that upright cycling positions are always more comfortable - although he is likely assuming people of normal range flexibility and other physical function.
Some writers object that their reach to the brake levers is impaired on the drops. Since the original set of answers were written, bike companies have put more effort into ergonomics. You can adjust the default lever position closer to the bar on almost all groupsets - I have personally done this on a current generation Tiagra groupset. Only Campagnolo may lack this feature on some of its groups (particularly 11s and earlier rim brake groups). To my recollection, SRAM may have been the first road groupset manufacturer to make lever reach adjustable. Additionally, Shimano and SRAM have the brake levers canted outward, which makes them easier to reach. I know that Shimano has put more effort into making the lever throws shorter (i.e. the distance the levers need to move to actuate a shift); I don't know specifically about Campagnolo and SRAM, but I assume they did so as well. These steps make the levers generally more ergonomic, but I suspect they most notably improve the situation in the drops.
While the OP didn't mention off-road cycling on bikes with drop bars, it's worth mentioning that the drops on many gravel handlebars are flared outward. The screenshot below is from PRO components' technical manual (a division of Shimano). The angle marked F in the top right diagram is flare.
Flared drops are a bit of a fashion trend, but they're limited to gravel bikes for now. Flare gives you some additional control while in the drops, and may help on technical off-road descents. Flare does not appear to be in fashion for road cycling, but if you benefit from it, don't be afraid to let your preferences dictate your handlebar choice.
In any case, if you are reaching too far forward when you are in the drops, you might consider seeking a handlebar with a shorter amount of reach, the definition of which is shown in the diagrams above. Handlebars also vary in their amount of drop (also shown above), so if you feel too low in the drops but not the hoods, changing handlebars might also address this.
For competitive riders, the case for the aero hoods position
Previously, I wrote about the aero hoods position. I mention it here for completeness. Your hands are on the hoods, but you crouch and shield your forearms from the wind. I find that I can rest my forearms on the ramps. This position has been shown to be more aerodynamic than the drops.
Professional road cyclists use this position frequently. With some practice, amateur cyclists can adapt to it, although my perception is that few people use it. Even more so than the drops, there are very few people who can spend really prolonged periods in this position. However, it offers a tactical option for competitive riders. You can place your wrists on the bar near the stem, as if riding a set of invisible aero bars, but this compromises your handling and it can't be recommended - although some pro road cyclists do it.
And last, as discussed on that post, you can mount aero bars. On long rides, many people use them merely for an additional hand position. Please exercise caution using them when riding in a group (preferably do not use them in a group, unless you are an experienced rider; more discussion in my other answer).
Miscellaneous modern handlebar terminology
I described flare above. Do not confuse with sweep, or S in the diagram. This usually refers to the ends of the handlebar being angled outward without changing the flare, and it is a feature on some gravel handlebars. To my knowledge, most gravel handlebars don't have the drops swept. The bars in PRO's diagram have zero sweep, and PRO only listed one handlebar with 2 degrees of sweep (i.e. insignificant).
Related, sometimes the tops of the handlebars curve backwards towards the rider, usually called backsweep. I believe that this is mainly a feature on flat bars, as Bikepacking.com shows on this page. However, if you consult the Nitto Noodle bar I linked earlier, the top is actually swept back slightly (by perhaps 3-4 degrees), and I believe some road bars may have slightly more backsweep. This reduces the bars' reach, but I don't believe the amounts of backsweep commonly seen are material. Alternatively, the Specialized Hover bar has tops that are actually swept upwards, which they call rise (measured in millimeters). I am not currently aware of any other drop handlebars with this feature, so there's probably not a standard term.