An Adapter has two frosted glass joints on either end, and is usually straight (some are angled) they serve to adapt frosted glass joints to/from any size or sex.

A Drop Down is the same as an adapter, but the frosted joints on either end will have a straight or curved tube of glass between them.  They are usually on an angle and will "drop" the height of the joint on your pipe.

They serve MANY different purposes!
First, they will distance the nail from your piece, which removes all of the heat from the joint on your pipe.
Second, they will serve as a "disposable" connector between your piece and nail.  They are MUCH cheaper than even our least expensive water pipes/ rigs!
Third use would be as a location for any oils that haven't vaporized fully to build up.  Instead of building up in the base of your pipe and getting nasty in the water, the drop down will catch most of it and keep it out of water!
They also will adapt to/from any size/sex joint.  That way you can use 1 nail for every pipe/rig instead of getting a bunch of different nails!
There are a lot more reasons, for example, personally I don't like to have the torch near to my water pipe at all, even though everything in science says it's fine to be close to it, I prefer to have distance away from the pipe itself.

If you read on any major nail brand website, they will all say the same thing - 

No matter what type of nail you're using, there is a lot of heat involved.  If you are perfect and can heat the top of your nail to an exact temperature every time, then you MIGHT be okay not getting an adapter, but for all of the humans out there it is almost a necessity! 

When you heat up the nail, there is heat that transfers down the nail.  If you're not careful (and even sometimes when you are!) the heat transferred can cause the nail you're using to crack the glass joint on the end of your water pipe or rig.  

Quartz, Ceramic, Titanium, or any combination in Male, Female, or Direct Inject can have the same issues.  

• A female nail goes over a male joint.  If the heat transfers it can expand the nail and actually slide down the frosted joint.  As the nail cools, the material will condense, effectively shrinking (a tiny bit) and can snap off all or part of the joint, or even break the whole downstem area.  
• A Male nail goes inside of a female joint.  If the heat transfers it can expand the nail inside and actually crack the joint or destroy the whole downstem as well.
• A Direct Inject nail goes inside of a male joint.  If the heat transfers (Takes a LOT more heat than the female or male) it is possible that the nail stem will expand inside the male joint and cause it to crack or break completely.

When using a glass adapter or drop down, it is still possible to heat up the nail too much and have it expand in or on the adapter/drop down.  But if the worst-case does happen, the only thing you would have to do is get a new adapter or drop down, instead of buying a whole new rig/water pipe.  Which cost less than a quarter of the price of even a "cheap" piece!!!

We wish you safe & happy smoking !!! :) 
Call our experts if you have any other questions!!!