Brown or gray depending on if you are using true complementary colors or not. If you mix cadmium red (red-orange) with phthalo green (blue-green) in a ratio of 3 to 1, you will get a dark gray. If you look on the color wheel, you will see that red-orange and blue-green are directly opposite each other so they are true complementary colors.
If you use the same cadmium red and mix it with sap green (yellow-green) you get brown. The color wheel shows that they are on color away from being complementary, thus the brown instead of gray.
Adding white to either of these will give you varying shades of gray or light brown. There are two related links which may be helpful. The second is a video demonstration.
All the above answers deal with pigments and paints. When dealing with light, the story is different. Red light added with green light in the right proportion makes yellow light.
The area of your eye that detects color is called the "cone". Your cones are specialized to send the messages for "red", "blue" and "yellow" to your brain. The absence of red is green, just like the absence of white is black. Thus, under normal conditions, there's no way to combine red and green. However, there are very specific conditions that cause your retinas to receive both a "red" and a "green" signal and send both signals to your brain, resulting in "reddish green". Doing so is difficult, one has to flash red and green in quick succession before the person trying to see reddish green while trying to keep that person's head very still so not to disrupt the illusion.
When you blend red light and green light of equal luminance, they cancel out. Since most forms of light contain RGY, most people see yellow rather than white when this "cancel" happens. The reason why green and red paint turns brown is because you've overloaded the paint with pigments, it's absorbed so much over every color that it can't display yellow or white, but your eye is canceling out either the red or the green (as well as the blue or the yellow) because you can't normally see both at once.