A respiratory droplet is an almost microscopic size bit of sputum. It you ever watch someone sneeze while observing them in sunlight you will see a spray of droplets exploding from their mouth and nose that you normally wouldn't see without this bright light. Those are the droplets they are referring to. When you have a cold, measles, TB, chickenpox and a number of other illnesses, they are transmitted this way. The pathogens are lurking in the sputum of the respiratory track, a great medium for bacterial and viral growth. So, when the person sneezes or their nose is running, the germs are riding on the droplets and spreading everywhere. An unknowing person who happens along and breathes in one or more of these infected droplets is likely to come down with the same illness unless they already have immunity or they are practicing precautions. And, you know kids. If they are sneezing or their nose is running, they are not paying attention to what they are doing with their sputum!
The difference between why you would use droplet precautions (or isolation) over airborne precautions (or isolation) or vice versa would be due to the size of the droplet involved. With droplet precautions (or isolation), the droplet is large and cannot travel very far. It cannot stay in the air very long and will alight on a surface very quickly. This is why the CDC guideline says a mask only need be worn when you are within 3 feet of the patient. Your danger of breathing a droplet in is only a concern when you are very close to the patient. With airborne precautions (or isolation), however, the droplets involved are much smaller and can remain suspended in the air for longer periods of time. The risk of infected droplets from a human source traveling through the air upon the ambient air currents is of greater risk and the likelihood of many other persons that the droplets could come into contact with breathing them in poses a much greater danger. If you read the CDC guidelines closely, they recommend that patients on airborne precautions (or isolation) be placed in rooms where the air is exchanged periodically or there is negative air pressure. This is to clear out the infected droplets that are hanging out in the air of the room. Many hospitals have specialized rooms for this kind of isolation where the air of the patient room is vented to the outside of the building rather than re-cycled through the entire facility via the normal heating/air conditioning environmental system. Every acute hospital I've worked in had at least one room on every floor that vented its air to the outside atmosphere. Whenever we had a patient with TB that needed to go into Respiratory Isolation (which is what it was called back then), we had to get the patient moved into one of these rooms.
I would say the terms isolation and precautions are interchangeable. When you see droplet or airborne isolation you should follow the CDC guidelines for droplet or airborne precautions as they are written: place the patient in a private room, wear a mask and also follow standard precautions. For airborne precautions the patient needs to specifically be in a negative pressure room, a room that vents its air to the outer atmosphere, or that recycles the air periodically. Remember that this also includes standard precautions. You can also see what standard precautions are (there is a link there on the webpage I listed for you). It includes gloving, gowning, special treating linen as if it is contaminated, using masks, face or eye protection and good handwashing.
I can't tell you which pathogens are classified into the airborne or droplet categories. They are classified according to the droplets they produce in the patient. My guess is that is has something to do with their ability to influence the production of mucus in the respiratory tract--in most cases, the mucus being more tenacious (thicker) and producing larger droplets. That, or the size of the pathogen adds to the weight of the droplet with certain pathogens being heavier than others. I just don't know which answer is correct. You'd have to depend on the information you got from your lecture. I would imagine your instructors are not going to go that deeply into the physiology of this. I would say that you are going to see airborne precautions (or isolation) more frequently in the medical units of a facility in patients with respiratory diseases and HIV who might also have TB or suspected of having TB and droplet precautions (or isolation) in Pediatrics where patients are more likely to have mumps, measles or rubella.
Hope that helps you out.