In the parlance of Talmud study and lomdus, what does the word "Hakira" mean and how should it be used? Why are Hakiras useful?
Literally it means "investigation" or something similar (and is used in other contexts, like that of evidence in court), but it's used to mean "hair-splitting".
Well, almost really. A chakira is an investigation as to the exact nature of something, and is usually stated as a binary choice: is X an A or is X a B? (X can be an object, a state, an action, etc.) A practical difference is generally sought so that the distinction is clear and known to be real, and, where possible, proof is brought as to whether X is in fact an A or a B.
It's useful in that it clarifies the nature of X.
UPDATE: msh210 beat me to it by a few seconds, but I'll post anyhow so you see the similarity between our answers.
A hakira seeks to suggest two subtly-different explanations for something. If explanation A is correct, then X should apply in some other case; if explanation B is correct, then Y should apply instead.
It helps to think of different ways to view the subject. For instance: