In their various parts of the world, all of these words mean sauce, at least some of the time.
They come from different cultures, though, and carry different connotations at least in US usage. Short answer, though: there are no absolute differences that you can count on.
This is a generic term in Spanish, and in South American cuisines. It can cover everything from a thick, dark mole, an adobo, a light and piquant salsa verde, or the typical tomato, onion and pepper sauce often served with chips in the US.
Some salsas are smooth; others are chunky. They can be cooked or raw.
Many people use the word to indicate the red, fresh or lightly cooked tomato salsa with onions, garlic, peppers and usually cilantro often served in Mexican restaurants in the US, but this is only the beginning of what salsas can be.
The generic term. Almost any flavorful liquid put on another food to enhance it. These range from elegently smooth (such as hollandaise) to quite chunky such as putanesca sauce.
Gravy tends to be a sauce made from meat drippings, and thickened to serve with the meat or its accompaniments.
In Italian-American usage, it often covers "Sunday Gravy" or a ragu made from tomatoes and one or more (or many meats) that in some communities was traditionally made on Sunday; and more generally, any sauce with meat in it.
I am told that in Italian in the context of pasta sauces, the actual word is (forgive my spelling) accompanimento, an accent for the pasta which is the star.
The word chutney derives from the India/Pakistan region, where again there are a myriad different sauces that carry the title.
In the US, it tends to be used for a chunky, acidic sauce made from fruit and/or vegetables. This overlaps considerably with jam or preserves, in this sense, although chutneys tend to be much more strongly spiced, and may contain complex combinations of complimentary ingredients.
Still, this doesn't cover the full range such as a chutney of pureed cilantro and/or mint, which is much more thin and sauce-like than the above more typical US usage.
You cannot really draw strict distinctions among these categories. You have to know what the specific sauce or item is that is being discussed to strongly draw inferences about it.