Base
A base is any part of a word that you can add inflections to, or that you can add prefixes/suffixes that change the meaning/part of speech. So "walk" is also a base, because it can have inflections (walking) and can be turned into different words walker is a noun. Walker is also a base, because you can modify it inflectional walkers is plural, and because it can have things added to derive new words dog-walker.
Stem
A stem is the form of a word that inflections get added onto. Most of the time this will be the root. "Walk" is the form that all the inflections grammar-affecting changes gets added to, when you add ~ing to it, it turns into a progressive verb or a gerund. ~ed turns it perfect. ~s makes it a plural noun, or makes it agree with a singular subject.
Root
A root is the part of a word that cannot be changed, and when added to creates different forms of the word:
"Walk" is a root, and can be changed in many ways: walking, walker, walkie-talkie, sidewalk, walk-light, walks etc. You will never have a word related to walking where the "walk" part gets changed, so it is a root.
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