Newton's theory of gravitation is a very close approximation to Einstein's general relativity, but it is measurably different in some cases (precession of Mercury, gravitational lensing, and more). Einstein showed that gravity can be neatly explained by the curvature of spacetime, that mass distorts the "fabric" of space (I use quotes because that's not the mathematical term for it, but it conjures a nice image that isn't too far off of reality). Objects move in straight lines along curved spacetime, but to us it looks like they go in loops around stars and such.

Special relativity has to do with the relation of space and time for objects sufficiently far away from each other that gravity doesn't affect them. Causality is enforced by this theory since nothing can go faster than light, and so all spacetime intervals we run into are time-like (That's just a fancy way of saying we only see wot's in our light cone).