Eubacteria are the so-called "true bacteria." They're distinct from archaea or archaebacteria, which form a separate kingdom, or domain, of life. Eubacteria are prokaryotes, which means they lack a nuclear membrane. Most have cell membranes and cell walls.
Bacteria with thick cells walls are termed gram-positive because they're susceptible to dying during a test called the Gram stain. The Gram stain is the first test used in bacterial classification. Bacteria with thin or absent cells walls are gram-negative because they do not trap the Gram stain dye.
Spherical bacteria are termed cocci; bacteria that form straight rods are called bacilli; and intermediate, short, fat rods are called coccobacilli -- and all can be gram-negative or gram-positive. Rigid, spiral-shaped bacteria are called spirilla and are only gram-negative. Flexible, independently mobile, spiral-shaped bacteria are called spirochetes and are gram-neutral. Finally, rigid, comma-shaped rods are called vibrios and are gram-negative. A few little-known and poorly understood bacteria have different shapes, such as the star-shaped stella and axe-shaped labrys. There are also two intermediate bacterial groups: Rickettsia, which are similar to viruses, have a variety of shapes, are gram-negative and can only survive inside other cells; Mycoplasma, which are similar to fungi, lack cell walls and include many species-specific, pneumonia-causing lung pathogens.
Cocci and bacilli are further classified by the linkages they form after cell division. Diplococci and diplobacilli stick together in pairs. Streptococci and streptobacilli form chains. Tetrad cocci stay in squares of four bacteria. Sarcinae cocci form eight-bacteria cubes and staphylococci form clusters.
Respiration, or metabolism, refers to whether a bacteria uses oxygen. Aerobic bacteria require oxygen, obligate anaerobes are poisoned by oxygen and facultative anaerobes can use oxygen or adapt to lack of oxygen.
If you have an unknown bacteria and you want to identify it, you'll typically perform a gram stain and then observe the colony appearance and the individual features. At that point, you can say you have, for example, a gram-negative, aerobic streptobacilli. You can then compare your sample to various known bacteria by placing it on different culture media that encourage the growth of some species and inhibit others, or by testing the sample for different known bacterial byproducts. As a final resort, DNA sequencing can determine if you have a known or unknown bacterial species or strain, provided you're comparing it to a species or strain whose genome has already been sequenced.