In the interests of node your homework, I've decided to enlighten everything2 on this obscure bit of high-school chemistry class nonsense. I learned this in the context of acid-base reactions (neutralization, titration, etc.), so that's how I'll explain it.
Acids and bases are never wholly concentrated- they must be placed in water for the chemicals to ionize (split up into their respective ionic parts). For example:
H2SO4 (sulfuric acid) ionizes in several stages. First, one H+ ion will separate to yield this:
H2SO4 --> HSO4- + H+
This is called the first ionization potential. Actually, the equation isn't quite correct since H2O should be over the arrow to show that that's the catalyst for the ionization, but hey- you try doing that in html. After this, another H+ will part company in the same way:
HSO4- --> SO4-2 + H+
This is the second ionization potential. Now the acid is broken all the way down into a sulfate ion and a hydrogen ion. In an ionic equation of any kind, this process is already assumed to have occurred, so the chemicals are already broken down into their ionic forms, like so:
Na2SO4 (s) + BaCl2 (s) --> 2Na+1 (aq) + SO4-2 (aq) + Ba+2 (aq) + 2Cl-1 (aq)
Again, there's really supposed to be a little H2O over the arrow. All those little things that look like this (aq) or this (s) signify the state of the chemical, which is important for finding out whether they ionize in water or not. (s) means the chemical is in solid state and (aq) means that it's in an aqueous state. When all these ionic parts react, we get a double replacement reaction that looks like this:
2Na+1 (aq) + SO4-2 (aq) + Ba+2 (aq) + 2Cl-1 (aq) --> BaSO4 (s) + 2Na+1 (aq) + 2Cl-1 (aq)
This is a nice little ionic equation. As you can see, Ba+2 (aq) and SO4-2(aq), both soluble in water, react to form to form BaSO4 (s), which is insoluble in water. To find out how a compound acts in solution (whether it's water, acid, or anything else) you have to look at a solubility chart, which can usually be found in the index of any chemistry textbook or on the internet.
In contrast to a run-of-the-mill ionic equation, the net ionic equation lists only those chemicals that actively participate in a chemical reaction. The chemicals you don't include in a net ionic equation are those that don't change during the reaction, for example:
2Na+1 (aq) + SO4-2 (aq) + Ba+2 (aq) + 2Cl-1 (aq) --> BaSO4 (s) + 2Na+1 (aq) + 2Cl-1 (aq)
The parts of the equation in bold are the ones you would not include in a net ionic equation, resulting in:
SO4-2 (aq) + Ba+2 (aq) --> BaSO4 (s)
This equation means that one sulfate ion and one barium ion in a solution would combine to result in solid barium sulfate, probably as a precipitate (solid stuff at the bottom of the liquid, like dirt seperating from water).
There you are- one more simple thing unnecessarily complicated.
SOURCES:
1. http://www.chem.vt.edu/RVGS/ACT/notes/Notes_on_Net_ionic_rxns.html
2. Castka, Joseph F., et al. Modern Chemistry. Orlando: Holt, Rinehart, and
Winston Inc., 1993. 444-445. | 
