Photobucket was first. They had the best free image hosting on the internet about five years ago, but like Ionbeam says, it is now advertising run amok. I even used to have a pro account there, but they have become essentially useless.
Full resolution images can be hosted in Imgur.com and it gives you a lot of alternative ways to link and post the images to a site, including easy to use thumbnail images of various sizes that let the user access the full size image by clicking on it, bit that doesn't put oversize images on forums or tax the data on a mobile device. No ads, just good hosting.
I still have my Photobucket account because I used it more than 10 years, but I am also looking for something different that works easily with a mobile app as well as with the laptop machine. Imjur is pending with their mobile app. If they get that, I'll drop Photobucket.