Film noir is a genre of movies from the 1940’s and 1950’s. These movies were pessimistic, gritty, and dark (noir is French for dark). The characters, even the protagonists, were often corrupt in some form, such as being alcoholics, lonely, or depressed. Common characters were corrupt policemen and corrupt politicians. The plot was often crime based, such as police or a private investigator trying to solve a crime.
The protagonist would often fail to accomplish his goals during the movies, sometimes thwarted by a femme fatale, or by a friend he trusted. The cinematography would be dark and menacing, for example, a dark and smoky alley in a big but empty city. Movies like Double Indemnity and the Malatese Falcon are examples of film noir.
Neo-noir is a modern recreation of those movies. It follows the same theme is a noir film, but was made after the 1950’s. Movies like Chinatown and LA Confidential are set in the time period of noir films (the 30’s, 40’s, and 50s), but were made outside of that time period. Blade Runner has the theme of a noir film, but set in the future. Another example is Who Framed Roger Rabbit, which is a noir film set in the 50’s, but has cartoon characters.
In short, if a movie has dark cinematography, corrupt protagonists, an unsolvable mystery or conflict, and if the villains win or at least force the “good guys” to do bad things, then it is noir.