The majority of the anatomy of a hookah does not vary by each hookah.
At the top of a hookah is the bowl, also called the head, which is where the shisha is placed. It is usually funnel-shaped and contains an area to put shisha. More information on bowls can be found here
The next part of the hookah, going down, is the tray. Trays are used to collect the ash from burning coals so the entire process of smoking is not a mess. Typical coals are made of pressed coconut shell. There are also quick-light coals, but the aftertaste from lighter fluid (or gunpowder, depending on the brand,) can cause hookah to be an unpleasant experience.
After the tray is the stem. The stem separates the bowl and the base, and it serves as an area to put hoses in the hookah. Hoses are used to pull the smoke from the shisha after it is passed through water at the base.
At the bottom of the hookah is the base. The base is usually filled 75 percent with water, which the smoke is filtered through when smoke is pulled into a hose.