The obvious differences in size and complexity between unicellular organisms such as bacteria and multicellular organisms such as animals obscure some more fundamental similarities. While a bacterium is a single cell that reproduces on its own and a large animal may consist of trillions of cells working together to reproduce, they are nonetheless both made of cells, and much of the functioning of those cells is shared between them. They all process nutrients, most of them breathe oxygen (anaerobic bacteria don't), and they all strive to survive and reproduce their DNA.
Multicellular organisms are all eukaryotes, while unicellular organisms can be either eukaryotes or prokaryotes. Eukaryote cells are much more complex, containing a nucleus and a variety of organelles such as mitochondria and lysosomes, while prokaryotes are much simpler and contain very little inner structure. In fact in many ways mitochondria themselves are basically prokaryotic organisms; they have their own DNA which is passed on differently. There is evidence that mitochondria actually emerged by ancient eukaryotic cells absorbing prokaryotic organisms into themselves.
Eukaryotic unicellular organisms are in some ways more similar to multicellular organisms than they are to prokaryotes; all the same basic cellular structures are already in place in a unicellular eukaryote such as the amoeba. To get from a prokaryote like E. coli to an amoeba you have to evolve a nucleus and a Golgi apparatus and transport proteins and so on; but put a whole bunch of amoeba together, and you sort of have a multicellular organism.
Indeed, that's basically what slime molds are; they straddle the line between unicellular and multicellular, consisting of many cells that join and split at different phases of their life cycle. But all these cells are basically the same.
The ancestor of all multicellular organisms was probably something like a slime mold; then at some point, hundreds of millions of years ago, it evolved the capacity to specialize its cells, making some perform different functions from others, despite having the same underlying DNA. (Think about all the different types of cells you have: brain cells, eye cells, skin cells, stomach cells, and so on; all of those cells share the same DNA, yet function quite differently.) That evolutionary innovation is possibly what made multicellular organisms as we know them possible.