The "style" itself doesn't have a specific name. However it is cartoony, it could be vector-based but you can achieve the same results with good old bitmaps, and it's quite saturated, colorful.
If your aim is to find strong references in order to build a similar yet original art direction, you may want to search in good graphic designer's portfolios.
There is no tough particularity to the way the environments are made. There is usually a single ground tile, a sky tile, one or two hills tiles, and then gameplay assets... mainly squares and rectangles for the structural elements, and characters.
Several plants or rocks are added to the first plane by hand in order to improve the overall composition.
Note that all the above applies for the original angry birds.