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.