I've heard the different places to put your hands on drop bars given different names, like "hoods" which I think is the flat part on top or maybe the brake levers, or "drops", referring to the lowest part of the bars. Can anyone tell me what the actual names of places to put your hands are?
"Middle of the bar", "On the tops", "Middle", "Top", "Tops", "Center"
Hands on the part of the bar closest to the stem. The middle has some width and you can slide your hands around there.
"On the corners", "Corners"
Hands at the outside end of the section where the bar turns. You can grip the corner different ways. There's a sort of position in between corner and hood ("ramp"?), too.
"On the hoods", "Hoods"
Hands on the brake hoods. The brake hood is that rubbery thing that the brake lever comes out of and on a modern road bike likely also contains the shifter. There's different ways to put your hands on the hoods, I'm showing two I use.
"On the drops", "In the drops", "Drops"
In the lower curved part of the bar. With the drops you can be leaned way down with your hands far forward/up in the hoods, you can be about where I'm showing in the picture, or you can be on the lowest part of the bar (where you can't reach the brake lever).
Note for all of these:
There are various different hand positions in each of those main positions, but not necessarily names for them. Some of the differences can be subtle and don't really look much different, but change the pressure points on your hands.
Most road-bike riders primarily use the hoods, and that's generally what's currently encouraged. It's the most comfortable position that you can brake and shift from and gives you good leverage. The drops are your fastest position and also give you good leverage. Middle is most comfortable for your hands, but gives you the least control.
I hate to simply copy+paste Sheldon, but...
Drop bars give you (normally) 4 hand positions: Middle of the bar...On the corners...On the hoods...On the drops.
I find that the middle is good when you need to breath more, the drops are good for aerodynamics, the hoods are for steep hills and I spend most time on the corners.