These are liquids that bear partial charges that can dissolve solutes.
Some atoms are better at attracting electrons than others. In an O-H bond, oxygen is better at pulling over electrons than hydrogen. So oxygen bears a partially negative charge and the hydrogen bears a partially positive charge. Thus a compound that has OH groups, like water and ethanol, are polar substances.
A solvent is a liquid that dissolve a solid. Water can dissolve table salt. Polar solvents can dissolve salts and other polar substances, like sugars and amino acids. But polar solvents cannot dissolve nonpolar solutes like fats. Only nonpolar solvents, like oil, can dissolve nonpolar solutes. What a solvent can dissolve depends on its polarity. Thus we when we discuss if a solvent is polar or not, we are trying to figure out what it can dissolve and what it cannot.
Polar: water, deuterium oxide (heavy water for NMR), ethanol, methanol, acetone, methyl ethyl ketone, isopropanol, n-propanol, acetonitrile, DMSO (dimethyl sulfoxide) or deuterated DMSO
polar solvents are all capable of forming hydrogen bonds with water to dissolve in water. Non-polar solvents can't form enough hydrogen bonding per molecule to dissolve in water.