The C language is mostly used for machine near programming today, like software drivers, smaller appliances and software that runs on small footprints (like home router or set top box). For web development you will find other languages like Java to be more common.
In either case, to become a good programmer you will benefit from learning both traditional and modern development languages, tools and methods. Learning about C as well as Java and Python is great. C++ or C# too etc. I'd recommend to learn programming in a broader sense for now and then later choose a specialization. But if C is your thing, for sure there will be jobs for you once you are up to speed.