All known living things descended from the same common ancestor, if you go back enough. My knowledge of the details is a bit sketchy, but I think the sequence goes something like this:
The first thing on the planet which we would unambiguously call “alive” was similar to what we would now call an archaebacterium. Originally, there was enough “food” free-floating in the soupy oceans that photosynthesis was unnecessary.
Some archaebacteria developed the trick of using sunlight and non-energy-providing ingredients to make sugar, from which they could get energy. These became what we would now call blue-green algae, and were the first photosynthetic organisms.
Some archaebacteria became more complex in ways I don’t fully understand, and became bacteria (not archae-).
Some of these bacteria developed internal “organelles” (roughly analogous to internal organs, but on a subcellular level), making them what’s called “eukaryotic”, and became the first protozoans. The interior of a prokaryotic cell like a bacterium or archaebacterium is largely uniform and undifferentiated: Anything the cell does can be done in any part of the cell. Some protozoans are photosynthetic, and some need to eat other living things. All are single-celled.
Incidentally, one way that organelles developed was by larger cells completely absorbing previously-independant smaller cells. The mitochondria (organelles responsible for metabolizing sugar) in almost all eukaryotes resemble bacteria, and have their own DNA completely independent of the rest of the cell. Another organelle of this type is the chloroplasts found in plants and photosynthetic protozoans, which are very similar to the blue-green algae. All of the photosynthesis in a protozoan or plant occurs in the chloroplasts.
All protozoans are single-celled, but some live loosly bound together into colonies. Some of those colonies became so dependent on each other that the individual cells couldn’t survive separately. At this point, they’re no longer considered many single-celled organisms, but one multicellular organisms. A multicellular organism that undergoes photosyntesis is a plant, while one which does not is an animal or a fungus (I won’t go into the difference between animals and fungi here).
From here, there’s a lot more development which occurs in the various lines, mostly geared around the development of specialized structures for particular tasks.
And I’ve never heard any definition of “life” which would include animals but not plants. You can’t even distinguish them by mobility: Some plants are more mobile than some animals. Sponges, for instance, are animals, but they spend their entire life anchored to a spot on the ocean floor, and can’t move any parts of themselves. By contrast, most plants can at least grow in particular directions, and in some cases can actively move already-grown parts. So you have sunflowers moving their head to follow the sun, venus flytraps snapping shut, etc.
I suppose you could say that, like other life forms, they simply react to external stimulus?)