Because the wave functions must be continuous, only integer numbers of wavelengths are allowed around the circle. That leads to a set of allowed wave functions that is discrete rather than continuous.
Don't take the shapes of the wave functions in these pictures too seriously: it's just a way to graphically show the periodicity. (More familiar looking graphs of these sines and cosines could be constructed by cutting the circles at some point and "unwrapping" them into flat planes.)
Recall that after setting c = 1, theorists measure time in units of length (like meters). When h-bar is set equal to 1, that implies that we will measure mass, momentum, and energy in units of 1 / length (for example, 1 / meters). This is considerably less intuitive, but it works well and helps to simplify the equations.
Finally, we combine these principles of quantum mechanics with those of special relativity to produce quantum field theory...
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