Animals and plants are both classified as Eukaryotes, and as such can form large, complex, multi-cellular organisms. There are several major differences at the cellular level that distinguish the 2 Kingdoms (Animalia and Plantae). Without getting technical, the most crucial difference in relation to your question is that plants contain chlorophyll, and as such can generate their own food using sunlight.
This is very important when you consider that plants (by definition, those Eukaryotes that contain chlorophyll) have therefore evolved to be very efficient at converting sunlight to 'energy', whereas animals have evolved without this option and have to eat what food they need. So they have instead evolved mechanisms that allow them to forage, or even to hunt, for their food.
At some point, a unicellular organism incorporated chlorophyll, leading to multicellular plants in the course of evolution. The ancestor of animals evolved independently from the last common ancestor of plants and animals. Therefore, plants and animals are quite diverged, e.g. in mating 'rituals' (in plants this involves flowers and seeds - whereas in animals there are mating and gestation periods), and systemic changes, such as the immune system and the circulatory system (these are generally much more complex in animals than plants). All of which has stemmed from the difference in evolutionary pressures applied by the choice of food-source.
As you have pointed out, there are exceptions to all rules, so not all plants have chlorophyll, but as a rule it is the case! (Wikipedia: "Some plants are parasitic and may not produce normal amounts of chlorophyll or photosynthesize.").