The term "antigravity" is really the stuff of science fiction rather than a term in common use in the scientific community. As such, it's rather ambiguous and could refer to multiple things. One idea pioneered in a story by H. G. Wells was a substance called "Cavorite," which would effectively block gravity's pull, allowing an object to lift away from Planet Earth (the tale was The First Men in the Moon). Other proposals have featured gyroscopes or electrodes or magnetic fields canceling out or altering gravity in a way similar way to Cavorite. Please note there has never been any realistic theory of any means by which a substance could block gravity (though under the right circumstances, magnetism really can counteract it). Under Einstein's General Relativity, gravity is not really a force at all--its sort of a side effect of the existence of matter/energy, which causes the structure of space to bend...well, more correctly, to compress. Any physical thing made of any form of known matter would merely add to that effect...
It so happens I did a previous post on the Higgs Boson in which I suggested that if it were possible to block the Higgs Field, you could wind up with an effect somewhat like the type of antigravity Wells imagined (because objects would loose mass), depending on details. Of course, as I understand the situation, there does not exist even a wildly hypothetical proposal for blocking the Higgs Field...so if conformity to what's possible in physics is a writer's goal, no story would be able to feature a "Higgs Field suppressor." If physics doesn't matter to you, hey, have fun...:)
Alternatively, antigravity has been imagined as a force that would work exactly the opposite the way gravity does--that is, it would repel all objects instead of attracting them. This idea in fact almost has a basis in theoretical physics. In the post previous to this one (on Realistic Warp Drive), I mentioned that something called "Negative Energy Density" would be required to create a warp drive field. So--if under General Relativity as mentioned above, gravity is the compression of space in response to the presence of mass/energy (really the energy is the important part--remember matter can be converted to energy via E=mc2), what would happen if you had an object that contained negative energy? Ignore for a second that "negative energy" seems to make no sense at all--IF you could produce such a state, negative energy would cause space to expand rather than contract, right? Yes, in fact it would.
Negative energy "antigravity" would thus allow you to expand space to make a tunnel to another region of space, a.k.a. a "wormhole." This kind of antigravity would also be required for warp drive, which would use regular gravity to contract space in front of the craft and "negative energy density" to expand space behind it.
An interesting side effect of an object with negative energy density is that it would be gravitationally drawn toward "normal" matter, because even though its gravitational force would be repulsive, its negative mass would respond by accelerating in the opposite of the direction of the force (which is a bit like if you filmed a car driving backwards and then replayed the film in reverse order--the car would appear to move forward). But normal matter would still be repulsed by matter with negative energy--so the negative energy density object would in effect "chase" the normal matter, never catching it if they were of equal mass, both continually accelerating faster (even though both accelerating, the total energy of the masses would cancel out, since the regular matter would gain regular energy and the negative matter would gain negative energy).
Yes, you could make a kind of engine out of matter with negative energy density, if you had any. The problem is, how in the world could anyone have such a thing as "negative energy density" or "negative energy"? That's all hokum, right?
Well, most likely. But there may be something more realistic that has a similar effect.
Antimatter, a very real substance with electrical charges opposite to normal matter, which has been manufactured in tiny quantities in high energy physics labs, most definitely has a positive energy density. When matter and antimatter collide they both instantly become energy--the total energy being equal to the mass (ran through E=mc2) of both the matter and antimatter.
However, certain attempts to harmonize quantum mechanics with relativity have considered that there must be a particle of some kind that actually does produce a force to bend space, in contrast to Einstein's understanding of gravity. Attempts to work out the mathematics for this particle, called a "graviton," also propose another presently unobserved particle, something called a "graviphoton." The graviphoton would produce another force, currently unconfirmed, that would be similar to gravity, only weaker and with a finite range. This force would add to gravity's pull on normal matter, but would repel antimatter.
So could such a force, one that repels antimatter, be employed in future technology? Future warp drives and the like?
The ability to repel antimatter certainly would come in handy if you're trying to keep the stuff from blowing up...the problem is of course that regular gravity would still apply so this repulsion force would be fighting against the ordinary gravitational attraction of matter (and antimatter). In my reading I couldn't find any reference to the idea that a graviphoton would cause space to expand--though since we're talking stories here and not physics, it could be imagined to do so.
It could be possible in theory to build a spacecraft or aircraft that produced all the effects we normally associate with antigravity...the problem is, you might have to make your craft out of antimatter...you know, just the most highly explosive substance in the known universe...perhaps a little trifle of a problem...
It certainly would be--though now I feel an urge to entitle a story, "Captain Elijah's Antimatter Chariot"...