Understand a Philosophical Argument That Keeps Coming Up

Most famous philosophical arguments exist in a simplified version that is widely shared and a real version that is more complicated and more interesting. The simplified version loses the problem the argument was designed to solve, which is the only way to understand whether the argument actually works. This takes the real version seriously, explains it without losing its substance, and shows where it succeeds and where it runs into genuine difficulties.

I keep encountering references to [philosophical argument or concept, e.g. Kant's categorical imperative, the trolley problem, Rawls's veil of ignorance, or Hume on causation] and I do not really understand it beyond the summary version. Explain it to me in a way that: starts with the problem it was trying to solve rather than the conclusion, gives the actual argument in plain English without losing its substance, shows where it succeeds and where it runs into genuine problems, and tells me one context where this argument is still actively used and one where it has been largely set aside. Do not simplify it into a slogan.