In a lot of the west, the most common rice is a long grain white rice, often exported from Mexico or China. I love basmati and wish it were more common here.
You've got some good advice there. White rice is just brown rice that has been milled of its bran hull. That takes away nutrients, but means a softer texture, milder taste and longer shelf life. People can have a preference for one of the other, and both have their uses.
Basmati, Jasmine and any other scented rices are very different animals that are typically used in pilafs and other strongly seasoned dishes. Magnificent things.
Wild rice is an entirely different animal. And a bit expensive. But absolutely lovely.
Long, medium and short grain rice, with no other characteristics noted, vary by their shape and their starch content. Long grain is your typical rice you'll find in a bowl at the chinese restaurant. Very good all-purpose, but you wouldn't try and make a risotto with it.
As for a real substitution, I don't know why your requirements are what they are, unless you've got a vitamin deficciency that rice bran helps out.
Long-grain brown rice is the most direct substitute given your requirements, but here's a possible alternative: parboiled rice
. It's been partially cooked before being hulled and polished. It has the flavor and texture of white rice, but has soaked up a good bit of the nutrients from the bran hull. If your requirement came under a doctor's advice, consult him/her first, but it might be an option because it's a nearly straight swap between that and long-grain white rice for flavor and texture, except it cooks a little bit faster.