The three most fundamental groups of organisms on the planet are

The three most fundamental groups of organisms on the planet are: a) bacteria, archaea, and eukarya. 

The Archaea possess the following characteristics:

1. Archaea are prokaryotic cells.
2. Unlike the Bacteria and the Eukarya, the Archaea have membranes composed of branched hydrocarbon chains (many also containing rings within the hydrocarbon chains) attached to glycerol by ether linkages . 
3. The cell walls of Archaea contain no peptidoglycan.
4. Archaea are not sensitive to some antibiotics that affect the Bacteria, but are sensitive to some antibiotics that affect the Eukarya.
5. Archaea contain rRNA that is unique to the Archaea as indicated by the presence molecular regions distinctly different from the rRNA of Bacteria and Eukarya.
6.  

The Bacteria possess the following characteristics:

1. Bacteria are prokaryotic cells.
2. Like the Eukarya, they have membranes composed of unbranched fatty acid chains attached to glycerol by ester linkages . 
3. The cell walls of Bacteria, unlike the Archaea and the Eukarya, contain peptidoglycan.
4. Bacteria are sensitive to traditional antibacterial antibiotics but are resistant to most antibiotics that affect Eukarya.
5. Bacteria contain rRNA that is unique to the Bacteria as indicated by the presence molecular regions distinctly different from the rRNA of Archaea and Eukarya.

The Eukarya (also spelled Eucarya) possess the following characteristics:

1. Eukarya have eukaryotic cells.
2. Like the Bacteria, they have membranes composed of unbranched fatty acid chains attached to glycerol by ester linkages . 
3. Not all Eukarya possess cells with a cell wall, but for those Eukarya having a cell wall, that wall contains no peptidoglycan.
4. Eukarya are resistant to traditional antibacterial antibiotics but are sensitive to most antibiotics that affect eukaryotic cells.
5. Eukarya contain rRNA that is unique to the Eukarya as indicated by the presence molecular regions distinctly different from the rRNA of Archaea and Bacteria.