I see some guitars with dark wood necks and others with light wood.
Does that have to do with the type of wood, like maple vs. rosewood vs. ebony or something? Or is it just the way they stain it?
Can you change the color of an existing fretboard?
totally dependant on the wood. maple is bright, creamy white naturally (usually) but sometimes more yellow. Vintage tint is often added to the clearcoat (maple is typically finished) to make it more yellow, or even a touch red. One horrifying extreme is, as frenchy put it, the "mcdonalds yellow" maple fretboard on the loomis.
Rosewood (Indian rosewood anyways, which is what you usually see) is brown, but widely varies. Often light brown, or dark brown. My main axe is an ibanez S which has more reddish tints to it.
Ebony is black, with some streaking. Sometimes its all black, but thats rare, and as is adressed in another thread, irresponsible to seek out exclusively. Easily died all the way black if theres a need.
That just covers the most common fretboards. The neck is 90% of the time maple or mahogany, which generally both get clearcoated, and often a colour applied at that phase. many typical SG's have a red clearcoat, which you can see the grain through.
Woods can often be stained as well, but its much more common on bodies than necks, cause stain can tend to come out with a lot of wear, and you clearly wear on the neck a lot more than on the body.