There are a large number of definitions for magic/magick/magik. I define it as the means and the process of altering reality to suit the Will. Aleister Crowley defined it, in Magic in Theory and Practice as "the Science and Art of causing Change to occur in conformity with Will." Dion Fortune defines it thusly: "Magic is the art of changing consciousness at will." (A quotation by William Butler in Apprenticed to Magic (1962) on p. 12 is the only original source I've traced for this.) Nevill Drury refered to it as "...the technique of harnessing the secret powers of Nature and seeking to influence events for one's own purpose." Isaac Bonewits, who favors not spelling it with a k, defines it as:

(1) A general term for arts, sciences, philosophies and technologies concerned with (a) understanding and using various altered states of consciousness within which it is possible to have access to and control over one’s psychic talents, and (b) the uses and abuses of those psychic talents to change interior and/or exterior realities. (2) A science and an art comprising a system of concepts and methods for the build-up of human emotions, altering the electrochemical balance of the metabolism, using associational techniques and devices to concentrate and focus this emotional energy, thus modulating the energies broadcast by the human body, usually to affect other energy patterns whether animate or inanimate, but occasionally to affect the personal energy pattern. (3) A collection of rule-of-thumb techniques designed to get one’s psychic talents to do more or less what one wants, more often than not, one hopes. It should be obvious that these are thaumaturgical definitions.
From Isaac Bonewits' website at http://www.neopagan.net/Pagan_Glossary.html
These are some of the clearer definitions that have been given; it is not hard to turn up many more, some of them extremely obtuse. In most modern usage, it is used to refer to their processes of reality alteration, not to be confused with stage magic. While some disagree with this, it is occasionally helpful to have the two spellings, particularly when one considers situations such as alt.magic versus alt.magick. It is notable that some of these definitions would encompass more "mundane" science and engineering.

Ok, how do you think it could possibly work?
Well, everyone disagrees. Some believe that it just changes the practicioner mentally in order to produce the desired results. Others feel it alters the energy one emits, therefore causing an alteration in the environment. Still others feel it works by getting in touch with supernatural entities and getting them to do things for you. Magick seems to follow a set of laws regardless of culture. Whether these laws say more about the truth of magic or the human mind is open for debate. It can run from affirmations and folk magick to ceremonial magick, with ritual and such. Guide to Spellcasting, by swankivy, gives a fairly good overview of the topic, biased towards a Wiccan ritual-based view.

Why with a k?
Aleister Crowley is believed the be the main source of the use of spelling magick with a "k" to distinguish it from stage magic/legerdemain. There are a few reasons, none of which can be proven to the satisfaction of all, as to why Crowley spelled it with a 'k'. First off, "magick" was the spelling in Elizabethan times, and was used in John Dee's diaries. This later fell into disuse in favor of the modern spelling by 1800 in England and by 1840 in the United States. Given that Crowley considered himself the reincarnation of Edward Kelly, this is one definite possibility. Another possibility is that since "K" is the eleventh letter of a few alphabets, and eleven is associated with the Qliphoth, which are the "underworld forces" that must be overcome in order to perform magick. A similar argument points to the ancient Egyptian khu, literally magic power, and relating to the vagina. Another claim suggests that based on Crowley's Magick in Theory and Practice, the spelling was changed to distinguish it not from stage magic, but rather from common Golden Dawn magic, which Crowley felt was bovine scatology.

This spelling change is apparently a written-only way of distinguishing the usage, as it should be pronounced in the same way. In fact, Mertseger argues (rather well) in magic that the difference is obvious by context and therefore that the spelling is moot. I have, however, heard people pronounce it mag"icik', basically repeating the sound at the end, as well as seen someone claim the pronounciation should be mage-ick.