What does it mean when something is said to "contain ions"? (I'm looking for a very basic level explanation because my only chemistry experience is one fast-paced high school course.) 

So, according to the professor of that course, ions are never found in nature as ions; everything would be neutrally charged. My question is, how can something have ions in it, without having them be neutralized? Doesn't nature want to lower potential energy, which would mean bonding the ions asap? Or am I misunderstanding things and said ions are already bonded to other ions, but they are still called ions even though the charge of the whole molecule should be neutral?
Your teacher is almost right but some terminology needs to be clarified.

Pure substances will almost always be electrically neutral (and the exceptions involve temporary local transfer of electrons from one substance to another as when you rub a rubber balloon on a wall surface). Beyond a certain level of charge separation you get enough electrical potential to cause sparks which allows the charges to equalise again. So persistent charge imbalance isn't easy to get. I think that is what your teacher means.

But if you look inside some pure substances, they do contain ions, just an equal number of each to avoid the bulk material having an overall charge. Common salt, for example, consists of ions (Na+ and Cl- in equal numbers). The substance is neutral, but the individual atoms making it up are all ions.

So your teacher is wrong if she meant that ions don't exist in nature but correct if she meant that bulk substances are electrically neutral.