Sometimes called the alternative to Photoshop (which is quite over-enthusiastic), GIMP is one of the most powerful free image manipulation tools for raster graphics. It is quite hard to master, however if you’re an artist it’s worth it.
It does not yet support non-destructive editing tools, these are planed in the next major version.

This can be directly compared to Paint.NET on Windows. It you’re looking for a quick and simple way to edit a picture, perhaps to crop it or add some simple details, I very much recommend this tool.

An excellent tool for artists that works great on touchscreens or with a drawing tablet. Works great for sketching, creating comics and according to their website also animating.
Being around since 1998 it matured greatly and can match most proprietary software.

The most popular vector graphics tool. It’s easy-yet-powerful and can allegedly match the capabilities of Adobe Illustrator.

Part of the Calligra Office Suite, Karbon is extremely capable and a viable choice if you either don’t feel comfortable with Inkscape or already use other Calligra tools.

If you specifically need a vector design tool that’s available anywhere perhaps check out Corel Vector. You can use it in a browser.
Unfortunately it is not Open-Source software.