It was formed in 25,053 BBY after the Unification Wars. A President of the Senate was named to control the government, though a Supreme Chancellor replaced that. It outlawed things like slavery, and smuggling, which caused a lot of worlds to draw away from it, since they supported those things. The Jedi Knights were developed to protect the Republic in the name of Peace, and Justice.
It had a massive army during the Clone Wars during called the GAR, which was essentially made of clones. It eventually won the Clone Wars, but the government was disbanded by Chancellor Palpatine, and remade it into the first Galactic Empire, with him as Emperor.