On a practical level (baseball, soccer, tennis, ...) Newton's account of gravity as a force with particular behaviors is entirely satisfactory. It is even used for launching satellites (well most of them), and predicting artillery impact points. It was this last application that made warfare even more horrendous than it had been (and safer for the artillerists, too). See Benjamin Robins, who was the most important early mover in the ballistic prediction business.
But, if something finds itself in a strong gravitational field (in Newtonian terms, Earth is not quite enough) or moving pretty fast (nothing in or around the Earth is fast enough**), Newton's account of his force is wrong. Mercury's orbit was famously intractable in Newtonian terms being just a bit off of Newtonian predictions no matter which brilliant calculator did the analysis. It turns out the the gravity is a force business is subtly wrong. In fact, gravity is intimately tied to the structure and geometry of space time, and is quite different near a mass than at some position well away from any mass. We will finesse the question of most of the mass in the Universe (dark matter, and probably dark energy) for these purposes just now. There are some ideas of what dark matter is (exotic particles usually) and some of these may be testable now that the LHC will be running at full design power for the first time.
Einstein's General Relativity (of 1915) made these connections tight enough for experimental test, the most famous of which as Eddington's eclipse expeditions in WWI. But Mercury's orbit matched GR predictions to well within observational error, and quite a few other GR predictions have also panned out. It's our best account at present.
So the obvious next question is why (how) does gravity affect the geometry of spacetime in the manner observed? Neither Newton nor Einstein were able to contribute much on this point. But a Universe pervading field (think ether, if you dare!? though the details are very different) for which a predicted particle would serve as the "force transmission" particle, was suggested by quite a few folks in the late 1960s. Higgs published first and so the field is named the Higgs field and the particle the Higgs particle. It has been quite hard to find. Fermilab should have had enough energy in its beam to find it -- but they didn't, and trawling through all that data they collected hasn't shown that they observed it but didn't notice. At least so far.
The LHC, being the most powerful ever built by a good factor, and after some teething problems, found something which is almost certainly the Higgs particle (though there may be perhaps 3 more in the family, as yet un glimpsed) in its first year of operation.
So, it appears that gravity is rather like the other fundamental particle forces (electromagnetic, strong, weak, and unified variants) in that the "force" is actually the exchange of particles which "carry" the force. There is much odd about gravity as these forces go, and much more work is needed to make sense of it. Perhaps Cavorite will be possible if we understand more?
The Higgs is popularly known as the "God particle" which is an egregious example of journalistic foolishness. One of the leaders in the search for the Higgs, for decades, was frustrated one day by its elusiveness (if it seixted at all) and was in the presence of some journalists, when he referred to it as that "Goddamned particle". The journalists (or one of them, stories vary), always on the alert for color in theoretical physics reporting, and knowing that they wouldn't get the entire phrase in their papers, just shortened it to "God particle". The physicist was not then and remained unhappy.
An exception to the necessity of taking into account GR on or near the Earth is the GPS satellite network. The timing requirements between the various satellites and a ground receiver are so exacting that the effects of GR on the orbits of those satellites is significant enough that a correction factor must be used in order to make the results accurate enough to use.