Both animals and plants are eukaryotes. They diverged from a eukaryotic ancestor, to give two branches.
From a microscopic point of view, the plant cells synthesize cellulose which is found in a particular structure, the pectocellulosic wall (or cell wall), outside the cell membrane (whereas the animal cells do not have this wall).Moreover, they have, in their cytoplasm, chloroplasts (endosymbiosis) that allow them to photosynthesize (even if certain parasitic plants have secondarily lost them).Animal cells also have special features: they synthesize molecules that are not found in plants, such as glycogen.
It also differs from a biochemical point of view, especially in the way of proclaiming energy.