Your heart is an amazing pump. It works reliably for decades, and it safely pumps blood -- one of the trickiest liquids around. In the same way, your blood vessels are pipes. They take the output from the pump and distribute it throughout the body. A blood pressure gauge is simply a way to measure the performance of the pump and the pipes.
There are two numbers in a blood pressure reading: systolic and diastolic. For example, a typical reading might be 120/80. When the doctor puts the cuff around your arm and pumps it up, what he/she is doing is cutting off the blood flow with the pressure exerted by the cuff. As the pressure in the cuff is released, blood starts flowing again and the doctor can hear the flow in the stethoscope. The number at which blood starts flowing (120) is the measure of the maximum output pressure of the heart (systolic reading). The doctor continues releasing the pressure on the cuff and listens until there is no sound. That number (80) indicates the pressure in the system when the heart is relaxed (diastolic reading).
If the numbers are too high, it means that the heart is having to work too hard because of restrictions in the pipes. Certain hormones, like adrenaline (which is released when you are under stress) cause certain blood vessels to constrict, and this raises your blood pressure -- if you are under constant stress, your blood pressure goes up, and it means that your heart has to work too hard. Other things that can increase the blood pressure include deposits in the pipes and a loss of elasticity as the blood vessels age.
High blood pressure can cause the heart to fail (from working too hard), or it can cause kidney failure (from too much pressure).