Bacteria eat carbon. Many different things have carbon in them; and some things can make their own carbon internally through a process that involves capturing light -- photosynthesis. Bacteria fall into both categories. Some make their own carbon-food; and others get the carbon-food by assimilating it from outside themselves. Bacteria that make their own are called autotrophs. Bacteria that get the carbon from outside themselves are called heterotrophs. This is one important distinction between bacterial types.
Besides dietary differences, bacteria have respiratory differences, too. Some bacteria require oxygen for cellular respiration. Other bacteria do very well without any oxygen at all. Oxygen-dependent bacteria are called aerobic; and non-oxygen-dependent bacteria are called anaerobic. Aerobic bacteria are those that are in action when you see food materials break down. If you are using fermentation, however, to make beer or sauerkraut or kimchee, the bacteria in action for those processes are anaerobic.
Sometimes the type of bacteria is determined by whether or not it stains with a special dye invented by Hans Gram, a Danish scientist. Because there are many bacteria that "stain" positive with Gram stain, and many that do not, this method has been a useful tool to quickly distinguish many similar-looking bacteria under a microscope. Although the Gram stain status explains nothing in itself about the functions of the bacteria, the negative or positive status of a bacterium allows doctors and scientists to quickly rule out many species of bacteria.
Below the kingdom in organic taxonomies comes phylum. Bacteria are part of the prokaryote kingdom, and that kingdom is broken down into 22 phyla. Those phyla include actinobacteria, aquificae, bacteroidetes, chlamydiae, chloroflexi, chrysiogenetes, cyanobacteria, deferribacteres, deinococcus, dictyoglomi, fibrobacteres, frimicutes, fusobacteria, gemmatimonadetes, nitrospirae, planctomycetes, proteobacteria, spirochaetes, synergistetes, tenericutes, thermodesulfobacteria and thermotogae.
Most bacteria exist in three basic shapes that scientists describe as coccus, bacillus and spiral. Coccus means round or oval-shaped. Bacillus means shaped like rods. Spiral is self-explanatory. There are rare but important exceptions to these basic shapes; but their rarity generally means the bacteria is immediately recognizable without any necessity for further differentiation, for example, by Gram staining. In addition to shape, observers can classify bacteria by the arrangement of those basic shapes. A diplococcus, for example, is a round bacteria that always appears in joined pairs.