VHS-C was a competitor of Video-8 in the camcorder market in the late 1980's into the 1990's, to avoid having huge camcorders that took full-sized VHS tapes.
Sony designed Video-8 from the bottom up as a new standard, and it gave better quality than standard VHS video tapes (memories of Betamax...). However, it was totally incompatible, and to copy your home videos onto your normal tapes, you had to run a cable from your camcorder into your VCR.
VHS-C stands for "VHS Compact" - which is an apt description of the setup. The cartridge itself was a similar size to a Video-8 cartridge, about 2.5 inches by 4 inches. However, the tape inside it was standard half-inch tape, exactly the same as runs inside a normal VHS cassette. Furthermore, the signals were recorded in exactly the same way as on a full sized VHS cassette. The main disadvantage of this, though, was that you only got about 30 minutes of recording on a VHS-C tape (or more with long play).
But this system had an advantage too. Rather than needing to run a wire from your camcorder into your VCR, VHS-C tapes fitted into a special adapter which looked distinctly like a normal VHS cassette with a big gap for the VHS-C cartridge. On insertion, the VHS-C tape was pulled out within the big adapter, so the adapter could then be inserted into a normal VCR and played directly.
This sounds great. But when it comes to the crunch, most people didn't want a whole load of VHS-C tapes as well as normal VHS tapes, so they ended up wanting to copy from VHS-C to VHS anyway. Which (unless you had two VCRs) had to be done in the same way as Video-8.
Of course, this is all pretty well history now with various Digital Video formats available for camcorders, and DVD recorders becoming more common at home.