What a fantastic question! Understanding general relativity in detail requires a graduate school educate (which is far more than I can give in an answer here!), but here are the fundamentals:
One of the key principles in physics is that physical laws should be the same in every reference frame; before the late nineteenth century, this was generally understood as implying that Newton's laws should be true in any reference frame moving at constant velocity. For example, if you're standing on the side of the road, and I'm sitting in a car moving in a straight line at constant speed, if you throw a ball in the air we should both be able to predict what path the ball will take (in our own reference frames) by using Newton's laws. This is called Galilean relativity.
Now, in the late nineteenth century, a physicist named Maxwell came up with a set of unified equations that describe electromagnetism. This was a revolutionary discovery, especially because since light is an electromagnetic wave, it became possible to calculate the speed of light from Maxwell's equations. The problem was that in order for Maxwell's equations to hold in any reference, frame, the speed of light had to be the same in any reference frame: this obviously did not obey the principle of Galilean relativity (imagine if you fired a beam of light as I drove by you in my car. According to Maxwell's equations, we would both measure the speed of that light to be exactly the same, even though our relative speed is different!). This problem puzzled physicists for a while, until in the late nineteenth and early twentieth centuries (a bit over a hundred years ago), they finally came up with a solution. The solution was Einstein's theory of special relativity, aided in part by the works of Lorentz and Minkowski. The key to this new theory was that space and time should be treated (mathematically) on the same footing; physical equations with this symmetry must be obeyed in any reference frame moving at constant velocity.
To get him started on general relativity, Einstein considered (or so the story goes) being in an elevator floating in space. As long as the elevator is not accelerating, someone inside it (you!) will feel weightless. However, if the elevator starts accelerating (say, by rocket engines), then as the elevator accelerates around you, you'll feel "pulled down" towards the floor. To you, it will look just as if gravity were suddenly turned on! This was Einstein's key insight: to the person inside the elevator, it is impossibly to tell the difference between the elevator accelerating and being in a gravitational field. Therefore, an observer in an accelerating reference frame should describe the same physics as if he were in a gravitational field: gravity can be described by being in a special reference frame! Using this idea, Einstein took special relativity one step further: instead of saying that physical laws should be obeyed by all reference frames moving at constant velocity, he said there should be physical laws that are obeyed by all reference frames, even those moving at non-constant velocities! When an equation is written in a form so that it is true in any reference frame (you on the side of the road, me in the car, or an astronaut in an accelerating spaceship), we say it's covariant.
Where does geometry come in? Well, in writing equations in covariant forms, time and space must be treated on the same footing. This means that we can picture time and space as a single object, a spacetime, which you can think of in terms of its geometry. Objects with no external force on them (we call these objects "freely falling") travel in "straight lines" in this geometry, but here, "straight line" doesn't mean what we might usually think of as straight: it means "line of least time." Freely falling objects travel along lines in spacetime that minimize what's called their proper time; these lines are called geodesics. What's cool is that if the spacetime is curved, then these geodesics do not appear to be straight; instead, they curve (think about the surface of a sphere, like the Earth: because the surface isn't flat, a "straight line" along it actually bends!)
Einstein's theory of special relativity says that mass and energy are what cause the curvature of spacetime, so if mass and energy curve spacetime, and curved spacetime makes objects travel on curved geodesics...then mass and energy makes objects move along curved paths! So, according to general relativity, the Earth orbits the sun not because the sun exerts a force on the Earth, but because the sun's mass bends the spacetime around it, and the Earth falls along a curved geodesic through that curved spacetime.
The idea is that space-time is a four-dimensional construct which is curved in the geometric sense: think for example of the surface of a ball, or the outside of any three-dimensional object, is itself a curved two-dimensional space, which has some different properties from your standard, flat, Euclidean space (e.g. parallel lines intersect, in a closed space like a ball, if you go far enough in any one direction you will wind up back where you started, etc.). The flatness of space is what causes space to behave in the Euclidean fashion like we normally think it does (e.g. parallel lines never meet), and at small scales space is flat: think for example of the surface of a frozen lake; it looks flat, but it actually follows the curvature of the surface of the Earth, which as you know is an approximate sphere.
In general relativity, the presence of energy/mass (which are the same thing in relativity, remember E = mc2) in the four-dimensional space time causes space-time to curve and bend. Objects when not being acted upon by a force follow geodesics in space-time - the equivalent to straight lines on a curved surface (such as any great circle on the surface of a sphere). The curving of space due to the presence of mass causes the geodesics to bend such that they come together toward the future location of the center of mass, causing massive objects to be drawn together, indeed all of space to be drawn together as you look forward in time. We observe this phenomenon as gravity, and the "force" exerted by gravitational fields is really just the inertia of the objects coasting along geodesics within them.
So goes the force-and-momentum description of gravitation in general relativity, at least. Most physicists however do consider gravity to be a force, because while acceleration due to a gravitational field may be an illusion, the fact that there is energy there and that gravitational fields can do work is not an illusion. You can argue that an object released some distance above the surface of the Earth, and which subsequently falls to the ground, is just moving along a geodesic and not acquiring kinetic energy while it is falling, once it hits the earth the relative velocities of the impact are transformed into heat, which most certainly IS kinetic energy. Work has been done. So in this sense, gravity most certainly IS a force, and physicists working on the grand theories of unification of the forces from what I understand do for the most part predict that at some point, even gravity will link up with the other three fundamental forces. I do not know why they think this; as I said, I am not a physicist. All of that said, their predictions are currently untested hypotheses, and could easily be wrong.
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