Why do cell membranes let small non-polar molecules through but won't let small polar molecules through?
The plasma membrane consists of hydrophobic and hydrophillic characteristics. Towards the outsides, they are hydrophillic, so they can create bonds with water. The insides are hydrophobic, allowing no water inside and keeping them tight together due to the polar forces.

An non-polar particle (if small), can pass through this because it does not interfere with the hydrophobic/hydrophillic (polar) nature of the plasma membrane. However, polar particles would not have the opportunity to move in, because the insides (hydrophobic) are literally afraid of water, or charges, don't allow polar substances to pass through.

So only hydrophobic (nonpolar), gases, and small particles (nonpolar) can pass through. There are exceptions of $H_2O$ passing through the membrane in small amounts because their electric charge is very minor.