Short answers, pulled from the story.
Einstein's static universe, also called the Einstein universe or Einstein static eternal universe, is a relativistic model of the cosmos that Albert Einstein proposed in 1917. It described a finite universe with spherical spatial curvature, assuming a uniform distribution of matter and no change over time.
Einstein introduced the cosmological constant to find a consistent solution to his field equations for a static universe containing a non-zero density of matter. Without it, the equations could not describe a universe that stayed the same size while still containing matter.
In Einstein's static universe, the cosmological constant Λ relates to the radius R and matter density ρ through Λ equals 1 over R squared, which also equals κ times ρ divided by 2. Here κ stands for the Einstein gravitational constant.
Einstein abandoned his static universe after Edwin Hubble discovered a linear relationship between the redshifts of galaxies and their distance in 1929. Einstein then proposed expanding models, including the Friedmann-Einstein universe and the Einstein-de Sitter universe, setting the cosmological constant to zero in both.
Many Einstein biographies claim he referred to the cosmological constant in later years as his "biggest blunder," but this is disputed. Astrophysicist Mario Livio has cast doubt on the claim, suggesting it may be exaggerated.
After abandoning his static model, Einstein proposed the Friedmann-Einstein universe and the Einstein-de Sitter universe, both allowing for expansion. In each, he set the cosmological constant to zero, calling it "no longer necessary... and theoretically unsatisfactory."