I'm not really sure how "in depth" of an answer you're interested in because it can get kind of complicated. I'm going to stick with the main idea, and if that's not enough, you can ask for clarification. Fair enough?
Anyhow, as you've already figured, your brain is made up of (among other gooey stuff) what is known as "grey matter" and "white matter." Matter is just another word for "stuff," so in essence they're saying the brain is made up of white stuff and dark stuff. Both groupings of "stuff" have different jobs to do.
The gray stuff is less than half of your brain, but it uses most of the oxygen you take in. That's because the gray stuff has the heavier chore to accomplish. It's made up of brain cells that send out the electrical signals that tell your body to run, pick your nose, see or smell. It's like lots of little command centers everywhere.
The white stuff acts more like a highway with one main office in the middle. It's the bigger part of your brain as a percentage, but doesn't require as much oxygen to do its job. The white matter is what connects the grey matter to the rest of the body and also allows it to connect to other pieces of itself. The other key feature of the white matter is that it controls, via various components, the body features that are involuntary, such as blood-pressure and breathing, heart rate, and certain hormones.
Both are required to go about your daily life, though in our society one wonders if many people are born without enough of one goo or another.