"A root may be defined as the simplest element common to all words in a related group [according to Paul R. Jenks]." In English, the words stature, stabilize, and stable all have "sta" as a root. This root is the same in Latin. Sta- words in Latin include statura, stabilis, and stabulum.
That part of the word that contains the essential meaning is called the stem. The stem may or may not be the same as the root. To make the noun "slaughter" (caedes) from the verb caedo to kill, you go to the stem = caed. To this stem, the inflectional endings are added. Suffixes are also added to the stem. If the stem ends in a vowel, the vowel may weaken or disappear.
The base is the part of the word that stays unchanged in inflection. Lingu is the base of lingua. It is common to learn conjugations and declensions in terms of the base and the endings that are added to it. When suddenly the base doesn't quite fit and changes before the ending, it's because you're not really using the base.
Source: A Manual of Latin Word Formation, by Paul R. Jenks, Boston 1912.