How does spatial curvature apply to the planets' orbits? We all know that in the presence of large, massive objects, spacetime is positively curved, more so the more massive it is. This means that the path of an object without any forces on it is a straight line, but through curved space, so the path of the object is curved. 
Does this mean the orbits of the planets are technically straight lines in curved spacetime? It sounds ridiculous, but the planets' motion is not acted upon by any forces.
 A: Yes, the orbits of the planets are simply geodesics in curved spacetime. A geodesic is the straightest possible curve in a curved geometry. (A 2D example is a great circle on the surface of the Earth.) Geodesics are “straight” in the sense that they “parallel transport” their tangent vectors along themselves. Parallel transport is a key concept in differential geometry. When you have a space with curvature, you have to carefully define how to compare a vector at one point with a vector at another point.
So gravity just becomes geometry, and objects free of other forces just move “straight ahead” in curved spacetime!
A: As G. Smith says, the worldlines of planets are indeed as straight as a line can be in the curved spacetime around the Sun. I would simply like to add the further clarification that this does not mean the spatial trajectories are geodesic, nor that the planets move in a straight line in a spatial sense. Rather, you should try to imagine the planet's trajectory extending in the temporal as well as spatial sense, winding around the Sun's worldline in a helix, but note that it is a very loosely wound helix. For Earth the trajectory extends one lightyear along the temporal axis during each year in which it goes once around the Sun---that's what I mean by loosely wound. Seeing this makes it easier to understand how it can be that no other nearby trajectory can find a route through the curved spacetime that is closer to 'straight ahead'.
If instead of considering spacetime you consider space, and plot the trajectory through space, then you do not get a geodesic (i.e. "straight" line) but rather an ellipse modified slightly into a sort of rosette shape. General relativity agrees with Newtonian gravity to first approximation, so it does not completely overturn the more familiar aspects of motion under gravity.
