If you are interested in a good basic overview, I suggest the On-Line Molecular Cell Biology at www.ncbi.nlm.nih.gov/books. Below is just a very brief and erratic introduction to the subject and several good links.

What is a membrane? Something like "a very thin film separating something from something else" usualy comes into one's mind.
In Latin membrana is the skin that covers the separate members of the body. It turns to be quite an accurate definition for the biological lipid membranes. Indeed, cells are very small "separate members of the body" covered by a skin made of lipids.

OK, after this short philological introduction we go straight to the subject. Everywhere below biological membranes composed of lipid bilayer and membrane proteins are referred as "the membranes" for simplicity.

Why are the membranes important?

• Every living cell is separated from the outer medium by at least one membrane.
• All cell compartments are separated from each other by membranes.

Therefore, membranes are the material basis underlying cell structure.
 

What do the membranes do?
 

• They separate one compartment of a living cell from the others
• They perform specific transport of chemical compounds
• They are involved in cell contacts and interactions
• They transduce signals into the cell
• They are involved in cell energy convertion

What do they consist of?

A fairly short answer will be: lipids and proteins.
 

• Lipids.

If we look into a  dictionary, we find something like :

    lipid : any oily organic compound insoluble in water
    but soluble in organic solvents; essential structural component
    of living cells (along with proteins and carbohydrates)

Let us expand a bit this succint definition.
 

• Lipid structure and function
• The World Of Membrane Lipids
• Lipid nomenclature
• Lipid Metabolism
• Structure of Fluid Lipid Bilayers
 
• Proteins.

Membrane proteins is actually what all the fuss on this website is about. They
determine all the specific functions of a membrane.
 
• Location.

Almost all membrane proteins are located...
YES! ... in membranes. It is accepted to think about membrane proteins as of particles
suspended in a two-dimentional liquid, a lipid lake. It is established that membrane proteins
are very mobile in the lipid bilayer. They drift and rotate.
 
• Common properties.

Membrane proteins are insoluble in water: they form aggregates, precipitates, etc. The cause of such
behaviour is hydrophobic properties of the protein. The most common structural motive of the membrane proteins is a transmembrane alpha-helix. The aminoacids composing the middle parts of such helices are mostly non-polar, while on the edges that protrude from the membrane there are polar and charged aminoacid residues.

• Special properties.

Most of the functions of any biological membrane is determined by the special properties of the membrane proteins integrated into it. Energy convertion, chemical, optical, thermal, and mechanical reception, signal transducing, photosynthesis (to name just a few) result from special properties of certain membrane proteins.
But here I will stop, for special properties of the membrane proteins is far too huge topic to cover here.

Links:
Membrane proteins of known structure
Molecular Cell Biology: Membrane proteins
Molecular Biology of the Cell: Membrane proteins