I have a video where some people speak with a loud background music. I am looking for a program that can increase the volume of the conversation with respect to the background music. If possible, free and compatible with Windows 7 or Linux.
It's not easily doable. You could try importing your soundtrack in an audio editor (Audacity, Sound Forge, etc), using the equalizer to cut down all frequencies except those that contain the most intelligible voice components (try starting around 600hz, 1kHz, 2.5kHz, 4kHz, add more treble to taste).
If you have an exact copy of the isolated background music (no dialogs), you can try importing it alongside the video's soundtrack, sync them so they play together, and revert the polarity on the isolated music track.