Wikipedia has a page detailing the history of star ratings. The film section reads:
In the 31 July 1928 issue of the New York Daily News, the newspaper's film critic Irene Thirer began grading movies on a scale of zero to three stars. Three stars meant 'excellent,' two 'good,' and one star meant 'mediocre.' And no stars at all 'means the picture's right bad,'" wrote Thirer. Carl Bialik speculates that this may have been the first time a film critic used a star-rating system to grade movies. "The one-star review of The Port of Missing Girls launched the star system, which the newspaper promised would be 'a permanent thing.
I don't know where you get the idea that star ratings are a thing of the past. Been to IMDb lately? Many critics, like the late Roger Ebert, prefer to use stars to quantify their opinion of a film.
How critics and viewers decide to give stars varies a lot and is often arbitrary or intuitive. Some decide on a rating based on their overall enjoyment of a film. Others might have separate ratings for aspects such as storyline, cinematography, screenplay, direction, etc. and average them out to arrive at a final value.
The arbitrariness of star ratings is pretty much why sites such as Rotten Tomatoes and Metacritic exist. They attempt to bring some statistical order to the chaos.