Bacteria cells are part of the prokaryote domain, whereas plants and animals are part of the eukaryotes. This the fundamental difference between them.
Prokaryotes (bacteria) are all unicellular organisms - the entire organism is just one cell. It has a membrane and cytoplasm, ribosomes for protein production, and DNA to encode the genetic information, but no nucleus or specialized organelles. Generally speaking, prokaryotes are pretty complex compared to their environments, but simple compared to the eukaryotes. They tend to live in more extreme, unstable environments, and have a lower overall metabolism (meaning that they don't produce as much ATP or protein).
Animals and plants both have a nucleus and specialized organelles, but their life strategies differ significantly, as reflected in their organelles. Animals have a more "offensive" approach, acquiring food by seeking it out and consuming it. This means that they prioritize mobility and energy output, so they are surrounded only by a membrane and have little in the way of bulky storage. They have only a few unique characteristics, such as lysosomes, which are small vacuoles used to break down material.
Plants have more unique features than animals; the most notable are the protective cell wall, the space-filling vacuole, and the photosynthesizing chloroplasts. These allow the plant to live more of a defensive life, producing its own food, and able to store excess nutrients for later, since they are incapable of seeking them out.