Bases, stems, and roots are the main components of words, just like cells, atoms, and protons are the main components of matter.
In linguistics, the words "roots" is the core of the word. It is the morpheme that comprises the most important part of the word. It is also the primary unit of the family of the same word. Keep in mind that the root is mono-morphemic, or made of just one "chunk", or morpheme. Without the root, the word would not have any meaning. If you take the root away, all that you have left is affixes either before or after it. Such affixes do not have a lexical meaning on their own.
An example of a root is the word "act".
Now let's look at what is a stem and a base and apply them to the root "act" so that you can see how they differ and interconnect to transform a lexical word altogether.
The stem occurs after affixes have been added to the root, for example:
Hence a stem is a form to which affixes (prefixes or suffixes) have been added. It is important to differentiate it from a root, because the root alone cannot be applied in discourse, whereas the stem exists precisely to be applied to discourse.
A base is the same as a root except that the root has no lexical meaning while the base does: "to act" is the infinitive of "act" and is structured with the base "act". In many words in our language, a word can be all three: root, base, and stem: "deer". The difference in their names lies on the way that they are applied during discourse (stem, base) and whether, on their own, they have any lexical meaning (stem, base) or no lexical meaning whatsoever (root).
An example of root, base and stem joined together is the word "refrigerator"
Latin root frīgerāre --> root; no meaning in English on its own; requires a change in spelling to affix suffixes
refrigerāre --> Latin prefix + root, with no meaning in English of its own yet
re- + friger + -ate + -tor--> prefix + root + suffixes that now produce lexical meaning = stem; spelling changes are required for suffixes.
The links included with the answer contain the Glossary of Linguistic Terminology for further information.