There's only two fundamentally different types of cells, and examples of single-celled organisms can be found in both types.
Prokaryotes have cells with no membrane-bound organelles. Basically, all the chemical metabolism of the cell happens out in the cytoplasm. All the genetic stuff, all the digestion, all the respiration, all that stuff happens all mixed up inside the cell, or along the cell membrane. These cells, however, can use proteins to form closed microcompartments to seperate a region inside themselves from the chemical environment of the cytoplasm.
Eukaryotes have membrane-bound organelles, like the nucleus, or Golgi bodies, or endoplasmic reticulum, that have specified functions. The mircocompartments of prokaryotic cells are a kind of evolutionary precursor to these structures. Eukaryotic cells also include various endosymbiotes like mitochondria and cholorplasts that handle various metabolic functions in exchange for shelter within the cell. These endosymbiotes have long since lost the ability to do anything but function metabolically but they still possess their own DNA, including vestigal sequences for the functions they no longer need to do.
The transition to multicellularity from a unicellular eukaryote is almost trivial, considering that a unicellular eukaryote is really a composite of several organisms already - itself, and the endosymbiotic mitochondria or cholorplasts. In the natural world we see a continuous range of organisms from basic unicellular organisms like the paramecium, to colonies of identical cells working together, even to organisms like sponges that are basically colonies of three or four different types of clonal cells.
Of course, that's not much different what a multi-cellular organism like you or I are - a colony of thousands of different types of cells, all clones of each other, working together but specialized for different functions, including reproduction.
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