What are the differences between hypersurfaces and manifolds?

A big issue here is how does your curve get its topology? Probably you are tacitly assuming the topology is inherited from the topology on the plane, i.e., the subspace topology.

But there are other ways to define a topology. We can present a curve parametrically so that it is homeomorphic to ${\bf R}$, but not in the subspace topology. Study this section of a wiki: Immersed curve. The image there is meant to show how this can happen.

Now let's return to your question at an intuitive level. If the topology of the curve in your question is inherited from the embedding plane, then it looks like no neighborhood of the origin is homeomorphic to an interval, so this would not be a one-dimensional manifold. Is the curve a hypersurface? I suspect you generated the curve by implicitly plotting an equation of the form $f(x,y)=0$, perhaps $y^2-x^4(1-x^2)=0$. If so, then yes--this would be a hypersurface. But I can envision defining the curve parametrically in such a way that the curve is a manifold homeomorphic to ${\bf R}$, but not in the subspace topology.