Replacement, aka displacement, reactions are those that involve two atoms or polyatomic ions "swapping places" in a molecule. For single replacement, the model is:
A + BC `->` AC + B
and for douple replacement the model is:
AB + CD `->` AD + CB
Hypothetically, you could see this happening in many situations with many different molecules, but most of the time the concept is taught in terms of ionic compounds dissociated in solution, since this is a faster and quantitatively more manageable approach. When combined with a table of solubility rules (linked below), replacement reactions can be used to create a key for identifying unknown compounds.
Solubility is a separate concept that would require further explanation, but in short, there are some things that dissolve and some that don't. Things that don't dissolve usually fall to the bottom of a solution and in that form are called precipitates. Replacement reactions that form precipitates are the best way of using these concepts to identify a molecule. For example; according to the solubility rules, many simple ionic compounds containing chlorine ions are soluble, with the exception of a few atoms like silver; thus, silver chloride (AgCl) would form a precipitate, and this would allow us to identify that either silver or chlorine was in the original pair of molecules if we made wise choices of what to pair up.
While we can identify some iodine precipitates, such as AgI, it appears that all potassium compounds are soluble; thus, we can use single and double replacement reactions to identify the iodine, but we cannot identify the potassium because there won't be any insoluble potassium compounds. The answers provided in this thread, such as using lead nitrate or silver nitrate, would produce identical results when combined with potassium iodide as they would when combined with other molecules such as sodium iodide. To positively identify the potassium, you would need to include other tests, such as taste, molality, or spectroscopic analysis.