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I was reading an article about the Oberth effect. It describes the orbital maneuver. The article said that the spacecraft performs a burn at perigee and gets into higher orbit.
So the question is "What will be the orbit of the spacecraft (or what will happen to it) if it burns at apogee?".
In general, there are six directions you can burn. You can burn along the path of the orbit (prograde), you can burn in the opposite direction (retrograde), you can burn at toward the center (radian-in) or away from the center (radial-out), you can burn normal to the orbits plane or in the opposite direction (anti-normal) and all combinations thereof.
Thus, a statement like "what happens if I burn at point X" is self-defeating, without stating in which direction the burn should happen.
In general, a burn prograde will never change the point of the orbit where the burn itself is performed. A prograde burn will raise the opposite point of the orbit, a retrograde burn lowers the opposite point of the orbit, while radial-in and radial-out change the shape (eccentricity) of the orbit and normal & anti-normal burns affect a plane change.
Thus, if you burn at periapsis in prograde direction, you raise the apoapsis. If you burn at periapsis in retrograde direction, you lower the apoapsis.
The Oberth effect says that burns at periapsis are most efficient. This is due to the fact that the vessel is faster at periapsis, and the fuel contains more stored kinetic energy.
To answer your question:
A prograde burn at apoapsis will raise the periapsis, a retrograde burn at apoapsis will lower the periapsis. But the delta-v per fuel mass will be (slightly) lower then when doing so at periapsis and utilizing the Oberth effect.
First rule of orbital dynamics: Whatever you do in orbit will affect the point at the opposite side of the orbit and leave the current point unaffected.
If you accelerate at the perigee, you change the position of the apogee. Respectively, if you accelerate at the apogee, you change the position of the perigee.
One well-known example of this is the Hohmann Transfer maneuver. This maneuver is used to change from a circular orbit to another circular orbit of a different height. When a spacecraft is on a circular orbit (apogee = perigee) and wants to reach a higher orbit, it will:
