The colour receptors (cones) in our eyes need more light than the ones that just register brightness (rods).
http://hyperphysics.phy-astr.gsu.edu/hbase/vision/rodcone.html
So for example, stars vary a lot in colour and e.g. some are yellow, some blue, some red, but to our eyes at a first glance if you aren't an astronomer then most of the stars in the sky seem white. If you look closely you may notice some have a red tinge (e.g. Betelgeuse which is a red giant, or Antares) and some may a bit blue-ish. If you could see them close up then they would be much more vivid in colour many of them. Just don't get enough light to trigger the colour receptors in our eyes strongly.
Similarly gas nebulae such as the Orion nebula - in the "sword" of Orion - should be reds and blues there but in small telescope or binoculars just looks white most likely.
The Aurora borealis similarly - for a much closer to Earth example - looks white when it is faint -you only see the greens, reds and blues when it is very bright. The colours are present all the time, but when the Aurora is faint, the colours are so dim that they don't trigger the colour receptors in our eyes so we see the Aurora as white although it is "really" coloured.
Then - many things in space also are grey. The moon rocks are basically shades of grey to our eyes. So is ice - and many things in our solar system are made of ice. Amongst the moons of the various planets, only Jupiter's moon Io is brightly coloured because of sulfur deposits on it. Others have interesting pastel shades e.g. Europa (also a moon of Jupiter) and Triton (moon of Neptune).
The planets are interestingly coloured to our eyes many of them so you have
Mercury - grey
Venus - pale yellowish
Earth - pale blue
Mars - red
Jupiter red / orange
Uranus - greenish-blue
Neptune - bright blue
See for instance this picture of neptune
http://en.wikipedia.org/wiki/Neptune
So there is a fair bit of colour in our solar system and in the universe, also a fair amount of grey things too - but because of limitations of our eyes in low light conditions then many things that are coloured will seem grey from a distance or in low light conditions.
Also - our eyes miss many things as the light we can see is very limited in its range.
To help there, to make details easy to see, then astronomers often use "false colours" to show distinctions our unaided eyes can't see (a bit like use of thermal imaging cameras at night).
So - some NASA photos will show things much as we would see with our naked eye. Some show them as we would see them if our colour receptors were much more sensitive than they are - so basically natural colour but light intensified. Some use false colour similar to thermal imaging.
And the universe is a colourful place even to our own eyes if you know where to look.