What background color should I use for an application that is the best for eyes?

Anything very light

It depends on your ganglia. In your eye there are cones that pick up red, green, yellow and blue light. This light is then transmitted to your colour opponent ganglion cells. Some are excited by red and inhibited by green; some the other way around. Some are excited by blue and inhibited by yellow; others vice versa. Phew! Everyone's brain is different. If you have a preponderance of ganglia that are excited by red, then red might appear more exciting to you than green. Or perhaps you have more yellow-excited than blue-excited ganglia - in that case blue might appear calming to you. Or perhaps colour processing in the brain doesn't actually work that way; there's lots of research but it's terribly complicated. We've figured out the physiology of the eye and quite a bit about visual processing, but since most of the interesting stuff happens in the brain we still have a lot to learn. So it's not really just your ganglia, although they play a part. You might also consider evolutionary psychology. Since we evolved to live in landscapes where much of the boring stuff was in a certain area of the spectrum, we have defined those (large) areas of colour as Green and Brown. Relatively small slices of the spectrum we have set aside for the colours of exciting things like ripe fruit and blood - we call those slices red, purple, blue, bright yellow. So you might consider green and brown to be less exciting colours, with green a more positive association because, you know, plants. Finally of course there are cultural associations for different colours, which will depend on your personal cultural makeup, and personal associations depending on your own life history. One useful thing though - you can certainly avoid overexciting your brain by using *less* colour. Reduce the saturation of the colour you choose, down towards grey. You can go completely grey if you like, but that might be more dull than your ideal. However a reduced colour (shifted towards grey) can look nice and calm and sophisticated. Brightness - towards black or white - is another axis you can explore. So, try a few things. Greens, greys, reduced blues are probably worth considering. Enjoy colour ^_^

You stated that you wanted a color that does not 'tire' the eyes; most of the responses above provide some insightful information on both the science of color and its practical application, especially vis a vis what 'excites' people to look at (excitement being a relative term). But for eye fatigue, I'm quite sure that IBM and others figured out a long time ago, that green text, on a black background seemed to cause minimal long-term eye fatigue. For those of us old enough to remember -- and have used Word Perfect, arguably the most innovative word processing program of its time -- pre-Windows; you'll remember that the default was green text on black background. Microsoft changed that with Windows, as the default background was white, and MS-Word (quite lousy at its launch) was black type on white background, which, though appearing similar to type-written text on a sheet of paper, was actually quite tedious on the eyes over time; but it has stuck nonetheless. But a great example of the contrast would be to look at an iPad and then a comparable text displayed on a nook or Kindle.  The iPad text may appear 'sexier', but the former will inevitably tire your eyes much more rapidly. Green on black works, but it isn't as pretty as it used to be (Windows 7 offers some default backgrounds that they call 'high contrast' and green on black is one of them. I had hoped it would be better. Anyway, based on all of this, if you're not looking for a 'sexy' look, I would recommend a dark background, with a medium-toned foreground for text -- bright enough to read, but not so bright that it hurts to look at.

Keep in mind that your monitor will show you different colors than others might see... so you should test background colors on your specific monitor... As you can see with the colors on this page: http://www.pagetutor.com/common/bgcolors1536.html Some are very easy to see and others are garish to our eyes.  What color font you have also plays into it... That's where a site like this comes in handy: http://www.pagetutor.com/colorpicker/index.html I personally like CCEEFF which I get by clicking on the top three rows... For no other reason... than it's fun to play with :-)

One of my biology professors a few years back always had his powerpoints with a deep blue background and white writing, which he claimed was much easier on the eyes than other typical color combinations, according to research. I don't know how accurate it it, nor have I seen research on the matter, but it definitely seemed to strain my eyes less than my other classes!

Use a dark color palette. Take a look at Adobe Lightroom if you need some inspiration, it has a nice UI. FWIW, most artists in look and lighting departments prefer dark palettes. My Linux terminals are all configured to use light text on a dark background.  IMO it looks prettier, too.