There are 3 local maxima and minima. These are the points that are the highest and lowest points respectively within a given region of the graph. In other words, they are the hill and valley points. In general, a shorthand way of saying "maxima and minima" is to simply say "extrema"
There are 2 inflection points. This is because there is a concave up region followed by a concave down region, then after that is a concave up region. Each change from concave down to up, or vice versa, introduces an inflection point in between.
There are 4 branches to this curve, so the degree is most likely 4. Even degree functions have both endpoints aimed in the same direction.