What are the main functions of the cell membrane?

The cell membrane is made out of two layers of phospholipids, a type of lipid with a head and two tails. These molecules' structure allows the membrane to be semi-permeable, meaning only certain molecules can cross the membrane. This is important as cells need to quickly gets things like oxygen and water, and get rid of wastes like carbon dioxide.

This image shows more of the membrane's structure:

You can see that there are also protein channels that allow materials to go in and out of the cell. These can transport molecules that cannot normally cross the phospholipid bilayer.

Molecules of the cell membrane are arranged in a sheet.

This membrane is called the fluid mosaic model as it is a mixture of phospholipids, cholesterol, proteins and carbohydrates.

Most of the membrane is composed of phospholipid molecules. These allow the membrane to be rather fluid.

Embedded in this membrane are proteins which give some structure to the membrane.

The 3rd components are proteins or glycolipids.

The membrane can seal itself if pierced by something very thin like a pin. But it will burst if it takes in too much water.

The proteins sort of float on the surface of the membrane like islands in the sea.

Cholesterol is also found in the membrane. It prevents lower temperatures from inhibiting the fluidity of the membrane and prevents higher temperatures from increasing fluidity.

The carbohydrates that are in plasma membranes are bound either to proteins or to the lipids. They form sites on the surface that allow the cells to recognize each other.
This is important because it all tells the immune system to determine whether a cell is foreign (non-self) or are body cells (self).