What substances can and cannot diffuse directly through the lipid bilayer?

It is very important for a cell to be able to exercise some control over what can enter and what can exit a cell through the cell membrane (lipid bilayer).  Because of the nature of the lipids that make up the membrane, only certain molecules can diffuse directly through the membrane down their concentration gradients.  The tails of the phospholipids that make up the bilayer are hydrophobic and non-polar.  Principle of "like dissolves like" applies here.  Substances that are unlike the lipids, that are hydrophilic or polar, will not diffuse directly through the membrane as they will be repelled by the lipids.  Also, in order to fit through the membrane, substances that can diffuse through the membrane must be small.  Only substances that are small, uncharged, and non-polar such as oxygen and carbon dioxide gases may diffuse directly through a lipid bilayer.  This process is known as simple diffusion.  

Other molecules like ions, amino acids, water, and sugars may also diffuse through the membrane, but they require carriers or channels in order to move down their concentration gradients.  These molecules may be too large, charged, or polar and a protein-based carrier or channel in a way shields these substances from the lipids that would otherwise repel such substances.  Diffusion through a carrier or channel is known as facilitated diffusion.