Love em or hate em they are here to stay. I'm a Gallente pilot so I'm personally rather fond of drones, but I completely understand to anyone out there that can't stand them. If you are new, and a Gallente pilot I highly encourage you to make tech 2 drones one of your top 5 priorities. If you're not a Gallente pilot I don't know when you should obtain them, but they definitely have their usefulness.
Currently I am have trained my primary character to use Gallente and Minmatar tech 2 light and medium drones. I highly encourage you to learn how to use more than one race of drones so that you can deal more than one type of damage. For the first month or two I only had hobgoblins and hammerheads and occasionally when I was running missions my drones would do awesome, and then the very next mission they were barely scratch the enemy. For a while I just figured they were stronger NPC's until I finally figured out that every NPC has a strength and a weakness to a specific type(s) of damage. Carrying more than one type of drone on you will help ensure that you never end up against someone that is resistant to your damage. Now I carry Hobgoblin II's and Warrior II's if I'm carrying light drones and Hammerhead II's and Valkyrie II's for medium drones. I use the Gallente version because they hit the hardest and the Minmatar version because they are the fastest.
Light scout drones should be used against Frigate and Destroyer class ships as they are fast enough to avoid most incoming DPS.
Medium scout drones should be used against cruiser and battlecruiser class ships since they hit harder and can still be difficult to target against.
Heavy drones should be used against battleships and up, although since I can't use heavy drones yet I use my medium drones on them and they work just fine.
Don't use medium or heavy drones on frigates class ships unless you want them blown up quickly, and when you get to tech 2 drones the cost to replace your drones starts to add up fast.
Salvage Drones are useful for helping to clean up a field after you've laid waste to it. I typically don't carry these drones on me unless I'm flying a drone boat like a Vexor, Myrmidon, or Dominix that can hold tons of drones. I'd rather fill the drone bay with more useful drones that I can use in a fight. I do load up my Noctis full of Salvage drones though, as they quickly help me clear a field while I tractor everything in and do some salvage as well.
Sleeper Salvage can be very difficult to salvage sometimes depending on the sleeper and size of the sleeper ship. Salvage drones will eventually be able to salvage a sleeper battleship but it may take 5-10 minutes sometimes as the chance is highly reduced. Definitely bring a personal salvager for these ships to save time.
Mining Drones do exactly that, mine. I have mining drones trained to 1 right now just so I can use them, but I don't ever bring them with me anymore. My alt that flies a retriever will load up his drone bay with hobgoblins instead to help with the rats that show up. If you have a large enough drone bay and drone bandwidth on a mining ship I can definitely see the use in launching 5 mining drones that also provides mining bonuses to drones, otherwise I leave these at home.
Eletronic Warfare Drones are something that I've been eyeing lately, trying to decide if I'm going to check them out. I see people try to use them, but they never tend to work as expected. I'm sure there is a good combination out there that is useful but I haven't figured it out yet.
Sentry Drones are likely my next type of drone that I will train for, but from what everyone tells me tech 1 sentry drones aren't that good, but tech 2 sentry drones are extremely deadly. The difference between sentry drones and other combat drones are that sentry drones are stationary and don't and won't fly after your target, however they have much longer ranges and can hit much harder. Tech 2 Sentry Drones are often used for POS bashing as they can add quite a bunch of DPS.
Logistic Drones are pure awesomeness in my opinion. Logi drones offer the ability to remote armor repair or remote shield repair your intended target, excluding yourself. I never use the light logi drones because they don't really offer enough repair ability but the medium shield and medium armor drones are extremely useful especially if you're flying a logistics ship. Typically if I'm flying an armor logistics ship like my exqueror i'll load up shield drones just to help supplement my ability to heal multiple damage types. My Logi drones never seem to draw aggro from NPC enemies but the logi ship controlling them most definitely will, so be prepared.
There are also fighter drones and fighter bomber drones that can be used on capital ships and up, and aside from knowing they hit like a ton of bricks, that's about as much as I know about them.
In summary, there are lots of different types of drones that have the ability to be very useful to you if you take the time to train them up properly. Most tech 1 drones help a little bit, but the tech 2 drones are just awesome, just don't forget to train up your other drone skills like Drone Interfacing that adds an additional 20% to damage for each level you train.