One of linguistic sub-studies is morphology. It is a study of the internal structure of words (Haspelmath, 2002: 1). In morphology, there are some things to talk about such as types of morphemes. There are actually many sources talking about it whereas I will give an overview of Katamba’s explanation.

1. Root, stem, base, and stem extender
According to Katamba (1994: 41), words have internal structure which is created by word-building elements. The elements include roots, affixes, stems, and bases.

 Root
Root is the irreducible core of a word, with absolutely nothing else attached to it. Roots can be free morphemes and bound morphemes. Roots which are capable of standing independently are free morphemes while bound morphemes are roots which are incapable of occurring in isolation (Katamba, 1994: 41).

The free morphemes include lexical morphemes and function words. Nouns, adjectives, verbs, prepositions and adverbs are parts of lexical morphemes, and articles, demonstratives, pronouns and conjunctions are parts of function words.

Although roots can be free morphemes, not all roots are free. Thus, the kinds of roots incapable of occurring in isolation are called bound morphemes. In English, the example of the morphemes is latinate affixes like –mit (as in permit, remit, commit, admit), -ceive (as in perceive, receive, conceive), pred- (as in predator, predatory, predation, depredate) and sed- (as in sedan, sedate, sedentary, sediment).

 Affixes
Affix is a morpheme which only occurs when attached to some other morpheme or morphemes such as a root or stem or base. There are three kinds of affixes which are explained as follows:
• Prefix is an affix attached before a root or stem or base like re-, un-, and in-.
• Suffix is an affix attached after a root or stem or base like –ly, -er, -ist, -s, -ing and –ed.
• Infix is an affix inserted into the root itself.

 Stems
Stem is part of word that is in existence before any inflectional affixes.

Example:
cats –> stem : cat, inflectional affix: -s
workers –> stem: worker, inflectional: -s

 Bases
Base is any unit whatsoever to which affixes of any kind, inflectional affixes and derivational affixes, can be added. All roots are bases. Bases are called stems only in the context of inflectional morphology.
Example:

Boys–> root: boy, stem: boy, base:boy, inflectional affix: -s
Boyish–>root: boy, base:boy, derivational affix: -ish
 Stem Extenders
Besides those elements, there is another kind of the structure building elements called stem extender.

In English, empty formatives are interposed between the root, base or stem and an affix (Katamba, 1994: 46). For example, in the formation of plural form of child (children) and breth (brethren), the irregular allomorph –en can only be added after the stem has been extended by attaching –r to child and breth.

2. Inflectional and derivational morphemes
Affix morphemes are divided into inflectional morphemes and derivational morphemes. They are bound morphemes. Those morphemes form words in different ways.
Derivational morphemes form words either by changing the meaning of the base to which they are attached and by changing the word-class that a base belongs to.
Example:
WordRoot
unkind–>root: kind, base: kind, derivational affix: -un (input: N, output: N)
Kindly–> root: kind, base: kind, derivational affix: -ly (input: Adj, output: Adv)

In contrast to derivational morphemes, inflectional morphemes form words either by not changing the meaning of the base to which they are attached or by not changing the word-class of the base belongs to.
Example:

Books–> root: book, base: book, inflectional affix: -s (input=output: N)
Walked–> root: walk, base: walk, inflectional affix:-ed (input=output: V)

Sources:
Haspelmath, Martin. 2002. Understanding Morphology. New York: Oxford University Press Inc.
Katamba, Francis. 1994. Modern Linguistics: Morphology. London: The Macmillan Press Ltd.