Describe the common methods used to identify bacteria

One way to identify bacteria is using Gram Staining. 
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The bacteria are stained with a purple dye (Crystal Violet). Though, most bacteria are then stained red. The bacteria are stained again with Potassium Iodide, then the bacteria are then washed with alcohol.
The bacteria that keep the dye after washing is known as Gram Positive bacteria. Likewise, those who don't keep the dye are Gram Negative bacteria. The bacteria are further stained with a pink dye (Safranin). Gram Negative bacteria will go pink after this dying, whereas Gram Positive bacteria will remain purple, from the original purple dye.

Then there is Ziehl-Neelsen Staining, also known as acid-fast staining. It's necessary as all mycobacteria have waxy coats on their cell walls that Gram Staining can't touch. All Mycobacteria test positive with this test, so, unfortunately, you cannot differentiate different species of Mycobacteria.