Drops freak the fuck out of people, including me. Even though I've done aerial silks for 7 years, I still make an instructor or classmate count "3 2 1" before I let go. Besides the fact that we're (understandably) innately loathe to willingly let ourselves fall, there's so much that can actually go wrong. You can miss your landing by a little or a lot, kick or hit something/someone that's a little too close, incorrectly estimate the amount of height you need, land too hard and get whiplash or strain something... well, before I scare you off, let me stop talking about what can go wrong and give you some tips on how to make it go right.
Land on your good side
Flips and drops are transitions. You're dramatically going from a starting pose to a finishing pose. Most students tackle a new drop by going into the starting position on the side they're most comfortable with. But what you should actually be thinking about is the side you're going to land on.
For example, say you want to fonji, and you're a righty. Your good shoulder inversion side is probably your right side. Your good brass monkey side is probably your right side, too. The problem is that in a fonji, you start and end on opposite sides of your body. The tendency of most students in this case is to start from their right side out of habit, and just hope for the best. Not smart! You have all the time in the world to set up your starting position, but only a split second to set up your landing position. That means your end pose has to be automatic, kinosthetically memorized. So work up your left shoulder inversion until you're ready to try it from there.
Don't forget to let go
The hardest part of dropping is letting go, especially with your hands. So some students attempt to skip that part. You can't skip that part. You will hurt yourself.
When I teach pole drops that involve letting go with your hands, I start by having students drill straight-up repels: just go into a shoulder inversion and spring off the pole onto the floor. (There are many variations, depending on what you're learning--and it's a cool move in and of itself if you can do it gracefully!) I don't want anybody trying to flip until they feel safe letting go.
There's a middle part, too
When first learning a drop, we tend to think of it as "point A, point B." Go from the start pose to the end pose. That's not dropping, that's teleporting. What we really care about in a drop or flip is the journey.
Sometimes this is mechanical--hit a star pose as you throw your side rotational drop in silks--and sometimes it's stylistic--sweep your outside leg dramatically as you're doing a shoulder inversion flip. But don't forget that what happens in between the point A and point B is the meat (or tofu) of the sandwich.
Who the hell are you trying to impress, anyways?
There are some really amazing drops out there. People are bouncing around all over the place, trying to make us shit ourselves by making it look like they're about to die. And it's awesome. But it's also dangerous. When you see a crazy flip and feel pressured to learn it, ask yourself: who the hell are you trying to impress, anyways? Is risking your neck worth impressing a couple other advanced polers? Is it worth getting like 200 hits on YouTube? 'Cause seriously, that what it comes down to--only a few people will know the difference. The general public won't think the flavor-of-the-month trick is any more impressive than an inverted crucifix nose dive. And you can put together gorgeous and dramatic routines without any flips or drops at all. So by all means do them if you want to. But don't feel like you have to, either.
Picture from http://es.tcktcktck.org/