Prepare for way more of my opinion on image hosts than anyone probably cares about xD I use tons for various purposes and feel like sharing.
~ The Bad ~
Photobucket went down the drain years ago, imo. The ads have gotten worse and worse, and so has the actual service. I used to use it almost exclusively but ugh. Its so terrible. I dread every Photobucket link someone gives me, generally art I purchased. .filetype links do not actually work as direct links and redirect you to the bogged down site full of Flash and garbage (ads etc), which is really gross, so I have to reupload the art to even save it to my computer.
I'm soooo glad more people are finally starting to abandon it.
Tinypic is just 100% bad news. They re-use URLs and your innocent pony photos can be replaced with NSFW/NSFL nudity or gore... I found this out the hard way. I used to use it because of the short URLs but Imgur covers that without deleting your files or replacing them with things you'd never post/look at. It also pretends to change file types on uploaded images which can be a pain to deal with. I have to upload Tinypic-hosted art I get to Imgur before I can save it to my computer/Dropbox or else it throws errors trying to crop it in Photoshop or re-upload it certain places.
Stash is really terrible. I would never, ever recommend it to anyone. Its super buggy and constantly forces you out of folders you're in, or even logs you out. It drives me nuts. I use it only as much as I have to to use dA. There are muuuuch better options out there.
~ The Good ~
For general photo uploading I use Flickr set to private (a lot of people see Flickr as a free-for-all to steal photos from). It works great, even if Yahoo screwed it up quite a bit and ruined a lot of organization. Its easy to post photos onto forums, especially ones with size restrictions since you can choose the size photos will be when you get the html/bbcode.
I use Imgur for all of my art uploading needs (like adoptables I get on the Arena or commissions I buy) or if I need an album of photos to show people, temporarily or long term. Imgur has basic editing and its worked well all the times I've used it. You can upload an unlimited number of things (I have over 10k) and an unlimited number of albums. One annoying thing is they recently changed it so uploading multiple files to imgur.com auto creates an album. If you go to username.imgur.com/all/ (or click username > Images in the top right) you can upload normally.
I've been using Dropbox for years to backup all of the art I get as well as special photos. I've had a folder for my son since he was born and upload and photos I get to it. Being able to share the album with people connected to their emails has been nice for me. I don't have to worry about them losing the link or it getting in the hands of someone I don't want it to. Like melodys_angel said its not really for hosting files (although you can totally do that) but its good at what it does. Most people I know use it for hosting huge files that need to stay fullsize or delivering odd file type commissions (.psd, .sai, etc) along with general backing up.
Recently I set up Google Photos (NOT Google+) as a backup. Its easy to make folders but if you install the auto-uploader it will not separate anything automatically. According to a Google employee friend that's on purpose - you're not "meant" to use the folders and should use the auto tagging. I thought that was dumb and have many folders for organization purposes anyways, haha. Uploading everything by hand was annoying when I was getting all the photos I've ever taken up but once I finished that it stopped being a concern.
Google Photos will auto sort everything by day/month/year (not connected to any albums) as well as auto sort by people, places, and things (in their own types of albums - one for every single person, place, or thing). You can search for any of this in the search bar. So for example any photo you've ever taken of a cat and uploaded will come up if you search "cat" and photos taken in 2012 come up if you search "2012". The tagging isn't perfect but its really good! You can make collages but when I tried there wasn't any variety with layouts and it was automatic so it didn't work for what I wanted (sale pictures). I don't know if there's any other editing abilities.
Lastly, I used Shutterfly to make a website for my collections. Its clunkier than other free website hosts (like Weebly, my preferred) but its worked well for this for all the years I've used it. Its not really a host either but if that kind of thing interests you its worth a shot. I assume there's some kind of photo editing built in since its primarily a place to order photos printed on stuff but I've never used it if there is.