Pointillism is a style of painting that, instead of strokes of paint, comprises paint dots. The style is supposed to give you the ability to mix the dots in such a way that you come away with the same representation of a naturalistic, traditionally painted depiction. It was at its peak in the last two decades of the 20th century, when it first started out.
Neo-Impressionism was an art movement of French origin started in 1886, characterized by painting that emphasized importance in optics. Its practitioners believed that separate, distinct application of pigment would produce better color vibrancy than simply blending it in together. This was referred to as mélange optique (optical mixture).
The greatest proponent of Neo-Impressionist color theory was Georges Seurat (1859 to 1891). Coincidentally, he is considered the founder of the Neo-Impressionist movement, and his scientific approach to painting led him to found pointillism. His crowing achievement---and the most famous example of pointillism---is "Sunday Afternoon on the Island of La Grande Jatte" (1884 to 1886) in which tiny paint dots of myriad colors used to depict several people in a park allows you to do the blending, as opposed to the painting already doing so.
If Seurat was the founder and most famous practitioner of Pointillism, Paul Signac (1863 to 1935) was its most ardent propagandist. He once stated that "the separated elements will be reconstituted into brilliantly colored lights." He himself did some Pointillist works, an example being "The Papal Palace, Avignon" (1900).
Henri-Edmond Cross (1856 to 1910) began his Pointillist period after meeting and spending time with Paul Signac. An example of work during that phase is "Cypresses at Cagnes" (1900). Cross was also associated with Seurat; he co-founded the Société des Artistes Indépendants (Society of Independent Artists) with him in 1884.
Art critics in the late 1880s coined the term "Pointillism" in an effort to ridicule the works of the artists who painted in this manner. Perhaps due to such critical reception, not too many artists of the age practiced Pointillism. Over the years, though, the term lost its sting, and today it is used to describe the paintings of Neo-Impressionist artists like Seurat, Signac and Cross without the negative connotation once associated with it.