There are two things that you can do to improve soundproofing. The easiest and least expensive method for soundproofing is to shoot some cellulose insulation between your drywall ceiling and your neighbor's floor joists. Cellulose insulation will cut down on the chatter of conversation as well as noise from the stereo and TV. If your neighbors have hardwood floors and a dog who scrapes his nails into the wood when he runs across the floor, then you will want something to protect against impact sounds. The best solution for someone living in a basement apartment is to install resilient bars. These are thin, Z-shaped bars of metal that attach to your ceiling on joists and against which you can attach a new drywall ceiling. The Z-bars are flexible and do not transmit impact noises. The insulation muffles the high frequencies, and the Z-bars the low frequencies. One last possibility that will definitely help is to ask your neighbors to lay a rug on their floor.