Being vulnerable increases bonuses when you make a contract and conversely increase the penalties for going light in your contract.  Vulnerability also affects strategies for a partnership bids and play.

Vulnerability:

Going Light Contract (undertricks)

If your contract goes down 1 trick (known as going light) and you are  not-vulnerable (not vul) you give your opponents 50 points per trick that you go light.

If you are vulnerable (vul) and go down the same 1 trick, you give your opponents 100 points ( -100 for your side). 

This is why you should be careful about overbidding when vulnerable.

Bidding & Making your contract

If you are not vulnerable (not vul) and bid and make "game" contract you will score a bonus of 250 points as well as the extra points for each trick won.

If you are vulnerable (vul) and bid and make "game" you will be awarded a bonus of 450 points as well as additional points for all the tricks you won.

Non-Vulnerable (not-vul) Under tricks penalties:

Down 1 = -50 points
Down 2 = -100 points
Down 3 = -150 points
Down 4 = -200 points

Vulnerable (vul) Under tricks penalty

Down 1 = -100 points
Down 2 = -200 points
Down 3 = -300 points
Down 4 = -400 points

Doubling

Doubling adds extra bonuses or penalties whether you are vulnerable or not, the difference is the amount of points you or the opposition are awarded.

You are doubled for penalties and you make your contract with overtricks!

Sometimes the partnership doubling for penalties gets it all wrong and the declarer not only makes the contract they make it with overtricks. For these overtricks when doubled massive amounts of bonus points are earned.
For each doubled overtrick, non vulnerable = 100 points
For each doubled overtrick, vulnerable = 200 points

How to determine which pair/s are Vulnerable or Not Vulnerable

West, North, East and South are all Vulnerable (red )

All positions  are Not-Vulnerable (grey)

North & South are Vulnerable; East & West are Non-Vulnerable

West & East are Vulnerable; North & South are Non-Vulnerable