Gravity
This article is only a brief description of the subject, and is not intended to give a full explanation.
Check out the "see also" or "references" sections, or Wikipedia's article for more detail.
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“”Stupid gravity!

—Homer Simpson 
Gravity, the opposite of Comedy (and not to be confused with gravy, which gravity keeps on one's food), has been speculated about since at least 800 BCE, when in the Indian subcontinent it was identified as the effect of weight. Every school of philosophers since that time has had some explanation for gravity  some approaching or even equaling that of Isaac Newton who in 1687 published Principia Mathematica, which laid down the inverse square law of attraction due to gravity.
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[edit] From Newton to Einstein
Since his day, Newton's Law of Gravity has proven sufficiently accurate to enable the prediction of the location of the planet Neptune. In the late 19th century, small irregularities in the orbit of the innermost planet, Mercury, could not be explained through Newtonian Gravity Theory, and it began to be questioned. Einstein's General Theory of Relativity solved the problem.
Everyone will be familiar with the visual and physical representation of gravitation as described by the General Theory, i.e., a stretched elastic membrane deflected by the weight of objects placed on it. This is a difficult visualisation of the mathematical idea of curvature in spacetime, requiring that the four dimensions of spacetime be visualised as a two dimensional elastic sheet which is distorting in some other dimension. Once this mental leap has been performed it is easy to see how varying masses have varying effects on the curvature of spacetime,; the larger the mass, the greater the distortion and the further reaching its effect.^{[1]} This is most often shown by placing a large, dense object (a bowling ball or cannonball) on the membrane and then rolling other balls past it at varying distances. As a thought experiment one can envisage fairly well the effects of the two objects on each other. The trouble with this demonstration is, of course, friction with the membrane.
[edit] Law of Universal Gravitation
The Law of Universal Gravitation, published in 1687 by Newton in his Principia Mathematica, is a theory which seeks to explain the force of attraction between all bodies of the universe. It is a theory, not fact, and as such the contents of this article are to be approached critically.
Mathematically, the law applied to two bodies is:
In which:
 F is the magnitude of the force between the bodies.
 G is the gravitational constant. In our universe, G = 6.67 × 10^{11} N m^{2} kg^{2}
 m_{1} and m_{2} are the masses of the bodies.
 r is the distance between the bodies, not the radius, don't forget that.
In words, the law states that the gravitational force between two objects is proportional to the mass of each object and inversely proportional to the square of the distance between them.
[edit] More advanced versions
The law of universal gravitation fails when confronted with situations involving strong gravitational effects. For example, it does not account for small perturbations in Mercury's orbit around the sun, or for gravitational lensing. General Relativity, which interprets the gravitational force as a consequence of warped spacetime, can accurately account for these observations, and has therefore surpassed Newton's theory as the preferred explanation for gravitation. However, the above equation is adequate for almost all practical calculations.
[edit] Quantum gravity
In order to discuss gravity and quantum field theory in the same mathematical environment, gravitation is theorised to be facilitated by the exchange of virtual particles called gravitons between masses. This is an extremely esoteric field and way beyond the scope of this article.
There should be waves of gravity created (analogous to bursts of electromagnetic energy) when massive events, such as the collapse of a star into a neutron star or black hole, occur. In an attempt to detect or measure these, there are several devices in place and under construction on the surface of the Earth which rely on the compression or expansion of physical objects at angles to each other. These are normally kilometerplus long tubes whose lengths are constantly monitored by laser reflection/interference. The anticipated difference in these lengths is of the order of 10^{10} cm or less. Scientists claim to have detected gravity waves in the background radiation from the Big Bang, indirect evidence previously existed from two pulsars spiraling inwards and losing energy. ^{[2]}
Gravity is one of the basic forces of the Universe, and the gravitational constant is thought to have remained the same since a vanishingly small time (the Planck time) following the big bang. As with the other forces, any variation, however small, would result in a very different universe.
The above has been extracted loosely from Wikipedia's "Gravitation" article which is a good start for a more indepth view of the subject.
[edit] Gravity is weak
Of the four fundamental forces, gravity, the strong force, electromagnetism and the weak force, gravity is by far the weakest.^{[3]} The only reason for its universal influence is that it is the only one of the four to not have a repulsive mode  it is always attractive. ("dark energy" might have some form of negative gravity which is causing the accelerated expansion of the universe)
[edit] Higgs boson
The Higgs boson, being the center of the mechanism for mass, has something to do with gravity. What, precisely, is a mystery to people outside the rarified atmosphere of particle physics.
[edit] Anti gravity of antimatter?
Until just recently not enough antimatter has been observable long enough to determine whether gravity repels or attracts anti matter, but in 2011 CERN’s Antihydrogen Laser Physics Apparatus (ALPHA) trapped 309 antihydrogen atoms for up to 1000 seconds. ^{[4]}
[edit] Controversy
See also intelligent falling.
There is no such thing as gravity : the earth sucks!^{[5]}
Also the Church of the Flying Spaghetti Monster has shown that gravity is only an illusion caused by the FSM in His goodness pushing everything downwards with His noodly appendage. From this it clearly follows that public schools should teach the controversy about gravity.
Some heretics, however, claim that everything is tied down by the hairs of the mane of the Invisible Pink Unicorn.
[edit] There really is no such thing as gravity!
Apparently it's just a phenomenon arising from other stuff.
“”Starting from first principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies. A relativistic generalization of the presented arguments directly leads to the Einstein equations. When space is emergent even Newton's law of inertia needs to be explained. The equivalence principle leads us to conclude that it is actually this law of inertia whose origin is entropic.

—Erik P. Verlinde 6 Jan '10 
... and they said quantum mechanics was hard to comprehend!
[edit] Quotations
“”Gravity is not a version of the truth. It is the truth. Anyone who doubts it is invited to jump out a tenthstory window.

—Richard Dawkins 
“”Christians don't believe in gravity!

—Peter Griffin, Family Guy, after Mel Gibson fell off Mount Rushmore 
[edit] See also
[edit] Footnotes
 ↑ Although technically gravity reaches from every object in the universe to every other, its effects due to very distant ones become much less noticeable.
 ↑ Einstein's ripples: Your guide to gravitational waves
 ↑ As a demonstration of relative strength, a small magnet, representing the electromagnetic force, can hold an object up against the gravitational attraction of the whole Earth.
 ↑ http://www.technologyreview.com/view/423901/antihydrogentrappedfor1000seconds
 ↑ And planes take off because they use special tarmac that stops the earth sucking.