I've never heard of brown meat. But there are two pairs of words like this.
1. red meat (beef) vs. white meat (chicken, turkey, etc.)
2. dark meat vs. white meat -- some parts of a chicken or turkey are darker / lighter than other parts. For example, the chicken breast is white meat. The meat closer to the leg bone is dark meat. This is only used to talk about different parts of a chicken or turkey, and maybe other kinds of poultry.
So part of a chicken can be "white meat" and "dark meat" at the same time. English is weird.