On the call-for-illustrations blog post at http://blog.protagonistlabs.com/post/87614173284/call-for-illustration-portfolios-from-artists, it says:

World banners (sometimes called cover banners) are 650px by 200px images. To support Retina displays and accommodate future uses of this art to promote or expand Storium, we’re commissioning these pieces at print-ready resolutions (300+ ppi).

Avatar images — including backgrounds — are 150px by 150px squares. We’re seeking print-resolution images here, too.

Card art displays either tall (100px by 160px) or wide (182px by 120px), depending on the card type. Again, we seek print-resolution images for these.

150x150 avatars at 300dpi are 1/2 inch on a side. Is that what you're looking for? 650x200 world banners would be 2 1/6" x 2/3" banners.

Print resolution shouldn't matter much if you've given the exact pixel size, though; anyone can run a photoshop batch script to adjust the dpi without otherwise affecting the images.

If you want print-resolution images that would appear in print as they do on the website--with the avatars being a bit bigger than 1.5" square on a 96dpi display--you may need to mention that you want "one-and-a-half inch square" avatars at 300dpi, which also look good at 150x150 pixels.

(On the one hand, asking for print resolution does cut a step out of later processing. On the other, it really is a very simple step--getting the exact pixel size correct is important; setting the resolution of all those images can be done in batch before any printing is involved.)