Short answers, pulled from the story.
The IS-LM model is a two-dimensional macroeconomic diagram used to teach the short-run relationship between interest rates and output. It plots a downward-sloping IS curve against an upward-sloping LM curve, and their intersection shows a combined equilibrium in the goods and money markets.
John R. Hicks introduced the completed IS-LM model by 1937, building on a paper he presented at a September 1936 Econometric Society conference in Oxford alongside Roy Harrod and James Meade. Alvin Hansen then extended Hicks' work into the version taught for decades.
Since the early 1990s, central banks have targeted inflation directly using an interest rate rule instead of targeting the money supply, which was one of the IS-LM model's core assumptions. This shift made the traditional model's money-supply story unrealistic, prompting David Romer and John B. Taylor to propose replacing the LM curve with a horizontal interest rate curve in 2000.
IS-MP is David Romer's 2000 proposal to replace the LM curve in the IS-LM model with a horizontal MP curve, where MP stands for monetary policy. John B. Taylor made a similar recommendation the same year, and many textbooks adopted this approach after 2000.
The IS-LM-NAC model is a 2016 variation introduced by Roger Farmer and Konstantin Platonov, where NAC stands for a no arbitrage condition between physical capital and financial assets. It allows high unemployment to become a permanent condition driven by pessimistic beliefs, rather than a temporary result of sticky wages and prices.
The IS-LM model is largely absent from macroeconomic research today, but it remains a backbone teaching tool in many undergraduate textbooks. Modern versions, such as Olivier Blanchard's IS-LM-PC model, have adapted it to reflect how central banks actually set interest rates.