Are all single-celled organisms Bacteria?
No, many unicellular organisms are not bacteria. Examples include (but certainly not limited to); some fungi, chlorella algae, and archaea. 

Bacteria are one of three domains in the classification of life. You can find more about the bacteria domain at the wikipedia page (It's a long and complex history which is hard to summarise here) and about the domains here. This last defines bacteria loosely consisting of "prokaryotic cells possessing primarily diacyl glycerol (DAG) diester lipids in their membranes and bacterial rRNA, no nuclear membrane, traditionally classified as bacteria." 

The composition of lipids in their membrane (DAG-esters) differentiates them from another major class of prokaryotes — Archaea, which have isoprenoid lipids in their membranes.

Initially Archaea was called Archaebacteria. Because of significant differences, including the membrane composition, they are now grouped into a different kingdom.