The differences between one-celled (or unicellular) and many-celled (or multicellular) organisms:
1) Nomenclature: Also called Prokaryotes also called Eukaryotes
2) Origin history first life forms on planet evolved much later
3) Example Bacteria, Archaea animals, plants, humans
4) Complexity very little, simple life form more complex life forms
5) Nucleus Absent Present
6) Membrane bound nucleus Absent Present
7) Mitochondria Absent Present
8) Nuclear Membrane Absent Present
These are some of the major difference between these two cell-classes. There are more differences,esp. due to presence of more complex molecules,tissues and organs in Eukaryotes, which make them more efficient as compared to prokaryotes.
For more differences and cell images kindly search for Prokaryotes and Eularyotes on your favorite search engine.
In addition to the above two answers, multicellular organisms show division of labour i.e. they have specialised tissues and cells for specific functions and processes. While in unicellular organisms like amoeba just one cell carries out all the basic processes of life like growth, metabolism, reproduction etc.
Unicellular organisms can survive without the help of other cells since their only cell is so complex. Multicellular organisms are much more complex since there are many cells that make up their body, and these cells are dependent on each other. Their cells are more specialised and thus different, while unicellular organisms have all functions in one cell. Multicellular organisms also depend on the health of all its cells to survive.