Unfortunately, since you do not own the space and cannot make deep changes to it, your options are limited. The general idea is to "decouple" the sound generation equipment from any rigid structural members, and to then isolate the air space that you are vibrating from any neighboring air spaces, so that the vibrations induced by the moving air into rigid surfaces is reduced to the minimum that is feasible to obtain.
Bass frequencies are going to be next to impossible to isolate; the best you will probably be able to do is to put your subwoofer on a folded towel, so that its contact with the ground is buffered and so the chassis can't induce as much vibration. Also try moving the subwoofer more than two feet away from any wall. this reduces the "coupling" effect that a subwoofer has with vertical surfaces, which sets them vibrating sympathetically (increasing the bass response as heard by you, but also transmitting it through structure).
Treble frequencies are much less penetrating, because their high frequency makes them easier to cancel out by their own reflection, even if that reflection doesn't produce a perfectly inverse wave. Basically, you will want to hang or install something very thick and substantial, but soft, on the wall that separates you from your neighbor. This soft material will literally absorb the sound by being compressible but not elastic; it absorbs the pressure wave without transmitting it through itself, so that the sound cannot contact and be conducted through the wall. Good candidates are polyester batting (i.e. quilts, insulating curtains), egg foam, acoustical panels, etc. Understand that these will also reduce the reverb time of the room, which will make the room sound "deader" and will make your stereo sound quieter as you will only get the sound waves emanating directly from the speakers instead of what's bouncing around the room.
Understand that if your neighbor wants it pin-drop quiet, your only real solution is to use headphones, or move. Sound pressure levels are measured on a logarithmic scale; an increase of 10dBA is a doubling of the perceived loudness, and a 10x increase in the energy contained in the sound waves. So, even if you were able to absorb 90% of the energy of the sound waves that are currently going through the wall to your neighbor (which is admirable, and will be expensive to achieve without tearing down walls), you'd only cut the volume as heard by your neighbor in half. If you were playing your stereo at a comfortably rockin' 85dBA, for the sound to be below your neighbor's threshold of hearing in a space with no perceptible ambient noise, you would have to block 99.999999% of the sound produced by your stereo. This is, I deem, impossible to achieve in an apartment built to standard multi-family residential code; to achieve this, you pretty much have to build an entirely new room within the existing room that is suspended on vibration-absorbing struts. This is NOT a project you undertake in a space you don't own outright and control down to the foundations.
