Magic is using your will to affect the world (or at least your personal piece of the world) and to help you sense things you normally don't sense.
Most of this stuff is stuff that can be done without magic. It's a way to focus your energies (physical or metaphysical, doesn't really matter).
Do people shoot fireballs or levitate or whatever? Nah, usually not. Mostly it's minor tricks, ways to speed up healing, adjust emotions, tap into things that your body knows but your mind doesn't.
Good example, water witching. Take two L shaped wires, hold the short ends in your hands, with the long ends pointing out. Walk over underground water (such as a water main) and the wires will cross. Your body knows the water is there (water has interesting electric properties, and you can use electronic devices to detect water also), so it's just a way to channel your bodies knowledge into something you can more easily observe (the wires crossing).
For the most part, magic is just unproven science. Astronomy was magic until we built telescopes big enough to confirm things. Psychiatry was definitely magic before it became an accepted science. Chemistry was magic before it became chemistry. Magic is just using science that hasn't become mainstream or proved yet. Some magic will turn out to be crap in the long run, and some will become a science eventually. The fact that most people that are into magic don't research it in scientific ways of course makes things a little bit more blurry.
Is worshiping a goddess needed in order to work magic? No, but it makes things easier for some (a calculator is not required to do math, but many will find it easier than doing it by hand). Is trapping it in all sorts of mysticism required to make magic work? Not in my personal opinion.
My personal belief... Magic exists. And if we understood everything about the universe, nothing would be considered 'magical', because magic is a moving target, it is that cutting edge of something, and if nothing is cutting edge, then there is no magic. I believe we are capable of much more than we realize. I think our bodies can perceive more about our environment than we realize, and I believe that we have more impact on the universe than what we can physically touch with our physical hands.
I'm a witch by some terms, meaning that many non-witch folks want to burn me at the stake. I am a warlock by some terms, meaning that many witch folks want to burn me at the stake. Either way, I think that magic exists, and that many forms of magic are complete crap, and I know I am wrong on some points. But mostly people want to burn me at the stake because I think that gods and goddess are crap too.