These are two types of intermolecular forces that have a great deal to do with the nature of materials. Read on..
Polarity can refer to either individual bonds within a molecule, or to the molecule itself as a whole. While we know that covalent bonding involves sharing of valence electrons between two atoms, it is generally true that if the atoms are different (i.e. have different electronegativity), the electrons will not be shared equally. Rather, they will be drawn toward the orbital of the more electronegative element. (This generally means the element further to the right and higher in a column of the Periodic table. You can look electronegativity values up for most elements.)
As a result, the atom that possesses the bonding electrons to the greater degree will show a partial negative charge, while the other atom is partially positive. This is a polar covalent bond.
A molecule is polar if, as a result of polar bonds, it shows, as a whole, one end that is partially positive while the opposite end is partially negative. The molecule is said to possess an electric dipole. Two molecules that are polar will exert fairly strong attractive forces on each other. This is important in dissolving for instance.
A hydrogen bond is a special case of a dipole interaction in which the H atom as a result of being bonded to a highly electronegative atom such as N, O or F, will be particularly positive in charge (although not an ion!). Now, another polar molecule having N, O or F atoms with partial negative charge will form a particularly strong interaction between the H of one molecule and the N, O or F of the other molecule. (The strength is due to the small size of N, O or F compared to the sizable negative charge they will carry, and the close proximity that occurs between it and the H atom.) This strong interaction is known as a hydrogen bond, and is vital to the formation of proteins and DNA for instance.