Without sodium, you would die. All cells in the body have electrical properties. Some cells actually depend very heavily on these electrical properties in order to function appropriately. Whenever a brain cell activates, it depends on a precise balance of sodium rushing into a cell and eventually being pumped back out. Oddly enough, when sodium rushes into a bunch of cells in a coordinated pattern, our brain is allowed to function as it should. The same thing holds true for muscles. Without sodium, your muscles would never have an opportunity to contract.
Sodium needs to be in a certain concentration in the blood and around your cells. "Concentration" refers to how much sodium is present in a given volume of fluid. For example, let's take 1 teaspoon of sodium and put it in 1 liter of water. It will taste somewhat salty. But if I double the amount of sodium and add 2 teaspoons, it will taste more salty because there is more sodium per any given drop of water. This is more concentrated.
If we now take 3 teaspoons of salt and put it in 50 liters of water, we may not even taste it! While there is more sodium, it is far less concentrated in the solution.
If we begin to mess around with sodium concentrations, then bad things will happen. Several years ago, a woman lost her life because she diluted the sodium in her body with way too much water. She drank way too much water and experienced something called hyponatremia. When this happened, sodium was not able to dance in and out of cells as it normally would and cells began to shut down.
Clearly, sodium is necessary to life. But as we know, if "a drop is good", it is not necessarily true that "a cup is better." More is not better. Balance is necessary.
From this foundational understanding of sodium's behavior in human physiology, we can begin to explore the problems with high sodium in our diets.