Short answers, pulled from the story.
Arthur Dempster, Nan Laird, and Donald Rubin published a paper in 1977 that gave the method its name. This work formalized an iterative process for finding maximum likelihood estimates in statistical models with unobserved latent variables.
Rolf Sundberg provided a detailed treatment of the EM method for exponential families in his thesis from 1971. His work built upon collaboration with Per Martin-Löf and Anders Martin-Löf at Stockholm University.
The first step is the Expectation step where one defines Q as the expected value of the log likelihood function. The second step is the Maximization step which finds parameters that maximize the quantity derived in the previous phase.
An EM iteration increases the observed data likelihood function but offers no guarantee of reaching a maximum likelihood estimator. For multimodal distributions the method may converge to a local maximum depending on starting values.
C.F. Jeff Wu published a correct convergence analysis in 1983 after noting flaws in the original 1977 work. Wu's proof established convergence outside of the exponential family as claimed by Dempster-Laird-Rubin.