Respect is a measure of how active you are within a territory. It affects the number of clients you'll get for the area as well as how high your prices can be. You primarily gain respect by performing drug deals in an area, but you can also gain it by using spray paint and giving out free samples in the area.
Respect drops over time. The rate that respect drops by is influenced by hideouts you own within the area, as well as the difficulties you have choose at the start of the game. The part of your respect bar that drops is displayed as the light blue portion of the bar.
There is a minimum level of respect which is maintained permanently by hideouts in the area, but also maintained temporarily by spray painting your logo around an area. This minimum respect level is represented by the dark blue bar in the picture above.
Finally, hideouts in an area provide a boost to your respect in that area.