How do I determine if a formula (ie. NO2) is an Atom, Ion, Molecule, Formula unit, or a polyatomic ion?

an atom is of a single element and is neutral

an ion has a charge

a polyatomic ion is made of more than 1 element and has a charge

a molecule is a covalent compound

a formula unit is one cell of an ionic compound.

If it is just one element with no other subscribes it is an Atom

If it is one element with either plus or minus next to it it is a Ion

If it is multiply elements with plus or minus signs next to it it is a Polyatomic Ion

If it is a metal bonded with a nonmetal it is Formula unit

If it is a nonmetal bonded to nonmetal it is a Molecule

If it is an element bonded to itself such as N2 and O2 it is a Diatomic Molecule

I think that covers anything if you have any other question just ask

You need to know the definition of all these terms in order to answer the question. (Don't you have a text?)

An atom is a single element.

An ion is a charged element (it has a plus or minus charge); in real life, it would generally be found as part of a compound with an ionic bond (one element "donated" an electron to the other element), but for some purposes we list it as if it were by itself. K+ for instance, is often how we refer to potassium. Either the questions will show a charge, or you will need to know which atoms typically carry a charge.

A molecule is a complete entity made of atoms joined by atomic bonds (they share electrons).

A polyatomic ion is two or more ions hanging out together (joined by ionic bond). Table salt, which can be depicted as NaCl or Na+Cl- is an example.

A formula unit is "The smallest repeating unit of a substance. The molecule for nonionic substances."
http://www.chemistry-dictionary.com/definition/formula+unit.php

Here are some helpful articles:
http://biology.clc.uc.edu/courses/bio104/atom-h2o.htm
http://www.emc.maricopa.edu/faculty/farabee/biobk/BioBookCHEM1.html

If it is atom, you can find it symbol in periodic table and it is without charge. Because by defination, the number of proton and electrons should be equal in number for an atom

For ion, it is formed by the addition of subtraction of electrons from or to an atom. Therefore, it will have charge (eg Ca^2+)

For fomula unit, from the defination from wikipedia. It is the lowest whole number ratio of ions represented in an ionic compound. Strickly speaking, it is not "a type of compound". For example, in a crystal of salt, there may be billion of Na+ and Cl- ions exist in the crystal. However, because the ratio of Na+ and Cl- is always 1:1. Therefore, we write NaCl instead of let say "Na10000Cl10000.

For polyatomic ions. From the name, we know that "poly" mean "more than one". Therefore, it is a combination of two or more atoms and contain charge.

For molecules. By defination, a moleules can exist by its own in atmosphere. It is different from ions. For ions, a cation (eg Na+) can be exist only if a counterion exist (eg Cl-). However, for a oxygen molecules for example (O2), It can exist by its own. A molecule usually does not have charge. Ionic compounds usually form lacttice structure but not molecules.
However, be careful that some atom can  be a molecule also. For noble gas, for an example, argon (Ar) can exist in atomsphere in its atomic form. Therefore it is an atom and molecule simultaneously.