Those large granular lymphocytes seen in blood smears that look quite different than the majority of the "small lymphocytes" are most likely Natural Killer or NK cells. It was originally thought that they were non-specific lymphocytes of the innate system that kill intracellularly infected cells and altered cells (i.e. tumor) cells, but it is now known that they have many more functions. Besides their cytolytic functions, it is now known that they have multiple receptors and play a role in self-tolerance. They are also thought to be capable of evolving memory cells, which could place them in adaptive immunity as well. The image depicts some of the functions of NK cells.
2. Vivier, Eric et al. “Innate or adaptive immunity? The example of natural killer cells.” Science (New York, N.Y.) vol. 331,6013 (2011): 44-9. doi:10.1126/science.1198687