That one was most likely done in Illustrator or some other vector based design application. However, Illustrator is very expensive so I'd Google "Illustrator Alternative" which should give you plenty of results. Pixelmator has SOME vector capabilities and will probably do everything you need it to do for around $30.00. Hope that helps!
The artist probably sketched it, scanned it, imported it into Adobe Photoshop to bump the contrast or clean it up. Then imported it into Adobe Illustrator and drew over the sketch using the pen and brush tools, then filled in the shapes with color (like the hair or skin). After exporting the file you can send it to a printer to print the shirt. Check out Zazzle.com for printing custom designed shirts.
By the way, GIMP is a free Adobe Photoshop alternative. Inkscape is a free Adobe Illustrator alternative. Although there are more alternatives in existence, please google around.