As others said, GIMP and Krita, especially on Linux.
If you want to try vector graphics, Inkscape
Clip Studio Paint is excellent and PaintTool Sai is preferred by anime artists.
I have tried all the free ones like GIMP e.t.c. they all really just suck compared to photoshop. Just download "Adobe Photoshop CS6". Its the last photoshop that worked offline so can be used for free once you get the installer from somewhere. Its excellent and only lacks some of the advanced tools that are in newer PS's. The layout is the same as the newest versions so you can easily follow any tutorial.
As mentioned above, there are several options such as Paint Tool Sai, clipstudio paint and the likes.
However, GIMP was also made for this kinda stuff.