Protic and Aprotic Solvents
For the comparison of solvents below, first remember from general chemistry that
H-bonds are not the covalent bond of an H with an electronegative element of the
same molecule. They are intermolecular bonds between an H covalently bonded to
an electronegative atom of the same molecule, and an F, O, or N of another
molecule. See my handout on Van der Waals forces.

In the paragraphs below, however, the other "molecule" can be an anionic
nucleophilic ion, such as F-.

Protic solvents: Protic solvents are those that are capable of hydrogen bonding which ultimately then means they have to have O-H or N-H bonds in their structure. The most common examples of protic solvents would be water, alcohols, or carboxylic acids which all have O-H bonds in their structures.
Aprotic solvents:Aprotic solvents are those that are not capable of hydrogen bonding. A notable example is acetone which has an oxygen and it has hydrogens but it doesn't have an O-H bond in its structure (the oxygen is only bonded to the carbon) and is therefore incapable of hydrogen bonding and is therefore aprotic. The other common aprotic solvents include ethers (like diethyl ether or THF), DMF (dimethylformamide), DMSO (dimethylsulfoxide) and CH3CN (acetonitrile).