Most historic schools of magic take one of 4 forms:
1) understand the natural order, and your place within it, so that you can guide its manifestations. (Various asian practices mostly like daoism, celtic druidism, etc.) Should be perfectly reconcilable with science, even though science in our world tends to show spiritualism as being non-real, since it isn't measurable or consistent. In a world where it does work, science should be able to collect data, and so it should be able to be reconciled.
2) "Like above, so below." Essential precept of sympathetic magic. It implies a fully interconnected state between essential actors, where discrete observances are just connected manifestations of the same event, viewed from different angles. Things like voodoo fall into this category. The voodoo doll is like the person being cursed, and if viewed correctly, is magically identical to the real object. Thus, damaging the doll, damages the person. Likewise, changes in the person should manifest in the doll. About the only thing close to this observed by science is quantum entanglement. As much as I hate to bring up game references, the mysticism created by micheal kirkbride for the TES series smacks HEAVILY of this, and has deep roots in gnosticism, which also focuses on this. At the deepest understanding, even the divinities can be moved by a well versed magic user, because they are bound by their natures, and manifestations the same as all other forces. Sympathetic magic is the most commonly practiced for of magic, and takes all kinds of shapes and cultural tones.
3) substitute the gestalt will of the natural order with your own. This is the basis of things like chaos magic. You imprint your will and desire onto the chaotic energy on which the universe is built, and the injected order of that will takes on a powerful but not entirely predictable manifestation in the higher level universe. Allistair crowley and his illuminated order ("the brotherhood of light") focused on this sort of practice, using dreams, personal sigils to represent complex ideas, and a little sympathetic magic to do the things he claims. This would bump heads with science pretty hard. The uncertainty principle especially. Not many cultures or orders use this brand of magic; in fact, most shun or forbid it outright. The exact processes by which one conveys their will into the embroiling chaos that gives rise to observable order is different between obervant practitioners, but usually involves the use of self hypnosis, followed by willfull supression of waking conciousness. This has been documented to have been done with a variety of techniques, including self inflicted pain, inducing a near death experience, and the use of powerful psychoactive drugs to cause a severely distorted state of conciousness. Due to the self-destructive, subertive, arrogant, and often selfish uses for this type of magic, many other magically minded philosophies consider this sort of practice to be particularly evil, with serious and far-reaching consequences.
4) divine magics, associated with the fullfillment of the divine will and purpose of a divinity. This is faith and belief based magic. Modern neopaganism is a mixture of this, natural order magic, and sympathetic magic. Kabalah is another, more pure form of this approach, with heavy trappings of sympathetic magic overtones.