There are a few common names for this raspberry relative: dwarf bramble, creeping raspberry and roughfruit berry. The latter name is a near-translation of the epithet lasiococcus, meaning “woolly-fruited” (via Calflora.net). Despite the fact that this taxon is named for its fruit, I haven't been able to find any photographs of its fruit online to demonstrate – perhaps a follow-up BPotD, if I can find a population of plants a little more accessible.
The genus Rubus is global in its distribution, occurring on every continent except Antarctica. How many species? Difficult to say. Here's an excerpt from a scientific paper on Rubus: “In eastern North America, the number of taxa in treatments of subg. Rubus section Rubus ranges from 240 species (Bailey, 1941–1945), to 198 species (Davis, 1990), to 12 species complexes (Gleason and Cronquist, 1991).” Source: Alice, LA and CS Campbell. 1999. Phylogeny of Rubus (Rosaceae) based on nuclear ribosomal DNA internal transcribed spacer region sequences. Am. J. Bot. 86:81-97. With such a wide range in numbers, it is safe to say Rubus illustrates that a single species concept has yet to be agreed upon.
Botany / photography resource link: Botany Photos, the impressive collection of plant photographs from a doctoral student in Finland, Maarten Christenhusz.