Why is methane nonpolar while methyl bromide is a polar molecule?
Polarity is a concept used to explain where the electrons are in a covalent bond between two elements. In essence, the nucleus of each element in a bond has a certain attraction to the electrons forming the bond between it and the other element.
If you have two identical elements the bond is perfectly covalent and completely non-polar.
But if you have two different elements, such as between H & Br, the bromine attracts the pair of shared electrons more strongly than the hydrogen and you have a polar bond. (If you looked at the relative position of the electrons between the two they would be shifted toward the bromine)
When considering if a molecule is polar or not, you also have to consider the three dimensional shape of the molecule. Carbon forms four tetrahedral bonds with other elements which are equally spaced around the central carbon atom. In the case of methane, you have four equal C-H bonds resulting in a nonpolar molecule. However, when you replace one of the hydrogens with a bromine, the situation has become unbalanced with a shift of the electron pair between the C & the Br shifting toward the Br. This shift results in the methyl bromide being a polar molecule.