Casper is a good ghost, who, much to the chargin of his uncles , prefers to help people, rather than scare them.
Not much is known about how Casper came to be a ghost, or how he ended up with the Ghostly Trio. According to the 1995 Casper movie, he died when contracted pneumonia from playing out in the cold for too long, but so far, that's the only version there is.
In the original Harveytoons animation, Casper has been seen standing next to a grave inscribed with "Casper", that supports the theory he is indeed a ghost. Originally, stories have shown that his visibility is due to wearing a sheet. In the Casper live movie, this is not shown and it is implied that he is naturally visible to humans as a white creature.
In some stories, Casper has claimed he is a ghost because his parents were ghosts, but he may have been lying about this to avoid talking about a painful death.
In a Harveytoons show where Casper takes care of a pair of kittens, it opens with seeing (or not seeing) him scrubbing himself off in the shower, completely invisible. He becomes visible by putting on something like a white cloth that people traditionally use to masquerade as ghosts. This implies that he is naturally invisible and puts this on so that he can be seen and interact with others.
In spite of this, he is apparently still able to move through solid objects without leaving this behind, so he may be able to extend his ability to move through things to other objects.
Being a ghost, Casper can become invisible and intangible, as well as fly.
He is also able to turn other things in close proximity (probably requiring to touch them) intangible as well. He does with with the sheet he wears to be seen, and also did it to feed a dog a bone when Casper scared off the cooked chicken that was going to be the watchdog's reward. It is unknown if he is able to turn living things intangible with him.
He has also been seen (such as in Which is Witch) fading into view to his standard white appearance from being invisible, rather than putting on the sheet. This indicates that he puts on the sheet for visibility, but is also able to turn it invisible to match his natural state. It is unknown if he is able to turn living things invisible too.
Casper seems to be able to selectively affect the world. His natural state seems to be to be intangible (along with making his sheet this way) since when he is unaware, things pass right through him. But he is also able to physically affect things, such as moving around things or people.
It is unknown if Casper has mass, he seems to be in sync with the earth's orbit, though not necessarily due to gravity since he is able to fly at will.