The best time to visit is in spring or fall. Paris has traditionally had an extremely temperate climate, and it never gets very hot or cold—although global warming has changed this dramatically over the past fifteen years or so. Anyway, the best weather in the city may be enjoyed during the long spring and fall. The spring season runs from roughly April to May; the fall season runs from roughly September to October. The normal weather is the same in both seasons, generally cool and sunny, with occasional clouds and occasional brief showers. Prior to April, the weather is usually a bit chilly, and after October, it tends to become a bit gray and rainy. Winter in Paris is not extremely cold (temperatures rarely drop significantly below freezing), but it can be dreary. Summer in Paris can be uncomfortably warm. If you are interested in the weather at this particular moment, CNN has a nice weather page on the city that is continuously updated.
The spring season is the most popular with tourists. The fall season has many of the same advantages, but without the tourists. Few people visit the city in winter, so if you come then, you‘ll encounter mostly natives. In July and especially August, many Parisians go on summer vacation, and the city is very quiet, with mostly only other tourists walking around in the summer heat. The above all presumes that the weather will be “normal,” in the sense of being the way it has always been. However, global warming is changing Paris weather so dramatically that you might want to consult the separate question on Paris climate.