Orbits in Strongly Curved Spacetime ..... this is really cool applet! ;D
http://www.fourmilab.ch/gravitation/orbits/

[color=blue]What about comets:[/color]
actually it is "trapped" in the potential well[img]http://www.fourmilab.ch/gravitation/orbits/figures/effpot.png[/img], so a comet is also obeying the planets' gravitational potential field.
Look closely at the satellite when it is very near Earth, it "spun" out further but only to come back eventually due to weak influence far away[img]http://www.fourmilab.ch/gravitation/orbits/figures/plorbit.png[/img].

[b][color=blue]My Understanding:[/color][/b]
Actually, i don't really know space-time curvature.

What i roughly know is
curved space-time [img]http://upload.wikimedia.org/wikipedia/commons/2/22/Spacetime_curvature.png[/img] explains why if the sun disappears immediately, it takes 8+ minutes for the gravitational potential of the loss of the SUN to affect the earth's motion because space and time is like this giant web connecting objects and events like (x,y,z,t).

http://en.wikipedia.org/wiki/Spacetime
In spacetime, a coordinate grid that spans the 3+1 dimensions locate "events" (rather than just points in space), so time is added as another dimension to the grid.