If the electronegativities of the atoms are different, the electrons are pulled to the more electronegative atom, making the bond polarized towards the electronegative atom (that is, the bond becomes "polar"). Electronegativity is a measure of how "attractive" that atom is to the electrons; the more electronegative atom gets a greater share of the electron cloud. The more polar a bond, the more like an ionic bond it is. Thus, bonds can be classified on a spectrum that runs between purely ionic bonds and purely covalent bonds, as in the figure shown below.

H2 is a non-polar covalent molecule (the electronegativities of the atoms are exactly equal), while HF is a very polar molecule (the electronegativity of hydrogen is about 2, while that of flourine is about 4, so the molecule is polarized towards flourine). NaF is so polar that we usually just call it ionic.
In a polar bond, the electrons in the bonding orbital are pulled towards the electronegative atom, and in response, the antibonding orbital becomes much spatially larger towards the less electronegative atom. You can see that in the diagram below for the bonding between hydrogen and chlorine.

The result is that it is easier to break a polar bond by donating electrons into the portion of the antibonding orbital on the less electronegative atom than it is to do that to the part of the antibonding orbital on the more electronegative atom.
One more important concept is that polar bonds are weaker than non-polar bonds. As the electronegativities of two atoms in a bond become more and more equal, the antibonding orbital's energy becomes higher and higher while the bonding orbital's energy becomes lower and lower, stabilizing the bond.
Polarity in a chemical bond creates partial positive and partial negative charges on the atoms involved (often symbolized as δ+ and δ–, respectively). Since the more electronegative atom has more of the electrons, it is more negative than the other atom, which tends to be more positive. This charge separation creates a "dipole moment", which is a way of measuring the polarity of a bond by its charge separation. The bigger the partial charges on the atoms, the larger the dipole moment. This is usually drawn as an arrow pointing towards the more electronegative atom, with the arrow starting from a plus sign at the less electronegative end, as in the picture below.
Molecules can be considered polar and non-polar as well, depending on the polarity of the bonds that form and the geometric arrangements of the bonds in the molecule. The CF bond in CF4 is very polarized towards flourine, but since the four bonds are symmetrically arranged in a tetrahedron in space around the carbon atom, the molecule as a whole is non-polar. You can think of it as the polarities all pointing in different directions and cancelling out. Meanwhile, because the bonds in H2O are arranged in a bent pattern in space, with an angle of about 110° between the OH bonds, there is an overall polarity to the molecule, because the polarities of the bonds don't fully cancel. Thus, water has an overall dipole moment for the molecule, whereas carbon tetraflouride does not. Water is polar, and carbon tetraflouride is not. 