There are roughly 60 elements that make up the composition of the human body.  Some elements are more prominent, with four that make up 96% of the human body's composition: oxygen, carbon, hydrogen, and nitrogen (in that order).  The oxygen and hydrogen bond and mostly appear as water (H2O) in the human body.

The other 4% is made up of seemingly random elements that do have specific functions for the body ranging from nutrition to metabolizing to maintaining the body's pH balance.  All of which are important in a functioning organism. 

Each element has a specific reason for being a part of the human body, or any other organism it appears in. 

Carbon- is found in all organic molecules, mainly because it easily bonds with itself along with the moderate amount of energy it takes to bond.  It is able to double and triple bond with itself when necessary and easily bonds with Hydrogen, which is also another element found in all organic molecules.

Hydrogen- also found in all organic molecules is known as the "glue" that holds together our DNA along with many other benefits to the body.  It has recently been found (or at least the news recently released) that scientists believe that our body stores Hydrogen in our muscles and it helps to regenerate cells, to regulate the water in the body, etc.  Hydrogen is very attracted to other ions of itself and thus makes a great "glue" for our elements.

Oxygen- is well known as needed for life.  It's what we breathe in and our body uses to supply nutrients to our body.

Nitrogen- is also found in many (but not all) organic molecules.  It is mostly found in the amino acids that make up proteins of our nucleic acids (that make up our DNA).

Calcium- is mostly found in the bones and in the teeth, however, the cause of osteoporosis is when there is not enough intake of calcium to support the muscles and the body thus takes the necessary calcium from the bones.

Phosphorus- is found in the nucleus of every cell in the body and is the primary energy source for the cells and is essential in nutrition for the nerves.

Potassium- is necessary for membrane function and nerve impulses and helps to eliminate toxins in the body.

There are many more that have a necessary function in the body, many that have to do with helping each element to fulfill its proper function.

The term "heavy metals" has been considered "meaningless and misleading" (according to IUPAC due to lack of scientific basis on the name).  What is mostly considered a "heavy metal" is five times the density of water.  The metals that are that have that mass are considered stable elements and thus do not easily form bonds with other elements and cannot be metabolized in the body.  Though some metals with a fairly light mass (Aluminum) are also considered toxic

Trace amounts of many "heavy metals" is necessary in the body, however there are some that are completely unnecessary in the bodily functions (toxic metals): Antimony, Barium, Beryllium, Aluminum*, Cadmium, Lead, Mercury, Osmium, Thallium, Vanadium, Thorium†, Uranium†, Radium†, Plutonium†, Americium†, Polonium†, and radioactive isotopes of other metallic elements.  These elements (some in trace amounts) greatly effect the human body's ability to metabolize food, and its basic functions. 

The radioactive metals can cause many different symptoms, including death, depending on the exposure level.  Anything from vomiting and nausea to cancer and coma. Acute Radiation Syndrome (ARS) can occur if the dose is high enough, it is a short term exposure, it penetrates to the organs and the person's entire body is exposed (the term nuclear meltdown may come to mind).  Immediate symptoms are nausea and vomiting followed by loss of appetite and energy, fever, diarrhea, seizures, coma, destruction of bone marrow and internal bleeding. It is possible that you will not die from ARS, however, if the symptoms do not end, the affected person will most likely die within a few months, most likely due to the destruction of the bone marrow.

*Aluminum is controversial in this list since those with proper kindey function can be exposed to huge amounts and have no ill effect.

†Radioactive