The dot is a Hindu, not Indian, tradition. It is a common error to confuse the terms Hindu and Indian, though. The vast majority of Indian people are Hindu, but India also encompasses many Christians, Muslims, Buddhists, and people of other faiths.
One of every six people in the world is Hindu, so it's not surprising the dot goes by different names in different dialects. You'll hear it referred to as a "tilaka," a "bottu," or a "bindi."
Traditionally, the dot carries no gender limitation: men as well as women wear it. The tradition of men's wearing the tilaka has faded in recent times, though, so nowadays you see a lot more women than men sporting one.
Red is the traditional color, but that's also changing — people now go with different colours depending upon their preferences. Traditionally, some unmarried women wore black dots, while married women sported red ones. Today, women often wear dots that match the color of their saris.
The position of the tilaka is standard: centre of the forehead. It represents a third, inner eye. Hindu tradition holds that all people have three eyes, the two outer ones used for seeing the outside world, the third one focusing inward toward God. As such, the red dot signifies piety as well as serving as a constant reminder to keep God in the front of a believer's thoughts.