The 3 Approaches

 There are three principal approaches to morphology and each tries to capture the distinctions above in different ways:

• Morpheme-based morphology, which makes use of an item-and-arrangement approach.
• Lexeme-based morphology, which normally makes use of an item-and-process approach.
• Word-based morphology, which normally makes use of a word-and-paradigm approach.

While the associations indicated between the concepts in each item in that list are very strong, they are not absolute.

Morpheme-based morphology

In morpheme-based morphology, word forms are analyzed as arrangements of morphemes. A morpheme is defined as the minimal meaningful unit of a language. In a word such as independently, the morphemes are said to be in-, depend, -ent, and ly; depend is the root and the other morphemes are, in this case, derivational affixes.
In words such as dogs, dog is the root and the -s is an inflectional morpheme. In its simplest and most naïve form, this way of analyzing word forms, called "item-and-arrangement", treats words as if they were made of morphemes put after each other ("concatenated") like beads on a string. More recent and sophisticated approaches, such as distributed morphology, seek to maintain the idea of the morpheme while accommodating non-concatenated, analogical, and other processes that have proven problematic for item-and-arrangement theories and similar approaches.
Morpheme-based morphology presumes three basic axioms:

• Baudoin’s "single morpheme" hypothesis: Roots and affixes have the same status as morphemes.
• Bloomfield’s "sign base" morpheme hypothesis: As morphemes, they are dualistic signs, since they have both (phonological) form and meaning.
• Bloomfield’s "lexical morpheme" hypothesis: morphemes, affixes and roots alike are stored in the lexicon.

Morpheme-based morphology comes in two flavours, one Bloomfieldian and one Hockettian.
For Bloomfield, the morpheme was the minimal form with meaning, but did not have meaning itself.
For Hockett, morphemes are "meaning elements", not "form elements". For him, there is a morpheme plural using allomorphs such as -s, -en and -ren. Within much morpheme-based morphological theory, the two views are mixed in unsystematic ways so a writer may refer to "the morpheme plural" and "the morpheme -s" in the same sentence.

Lexeme-based morphology

Lexeme-based morphology usually takes what is called an item-and-process approach. Instead of analyzing a word form as a set of morphemes arranged in sequence, a word form is said to be the result of applying rules that alter a word-form or stem in order to produce a new one. An inflectional rule takes a stem, changes it as is required by the rule, and outputs a word form; a derivational rule takes a stem, changes it as per its own requirements, and outputs a derived stem; a compounding rule takes word forms, and similarly outputs a compound stem.

Word-based morphology

Word-based morphology is (usually) a word-and-paradigm approach. The theory takes paradigms as a central notion. Instead of stating rules to combine morphemes into word forms or to generate word forms from stems, word-based morphology states generalizations that hold between the forms of inflectional paradigms. The major point behind this approach is that many such generalizations are hard to state with either of the other approaches. The examples are usually drawn from fusional languages, where a given "piece" of a word, which a morpheme-based theory would call an inflectional morpheme, corresponds to a combination of grammatical categories, for example, "third-person plural". 
Morpheme-based theories usually have no problems with this situation since one says that a given morpheme has two categories. Item-and-process theories, on the other hand, often break down in cases like these because they all too often assume that there will be two separate rules here, one for third person, and the other for plural, but the distinction between them turns out to be artificial. The approaches treat these as whole words that are related to each other by analogical rules. Words can be categorized based on the pattern they fit into. This applies both to existing words and to new ones. Application of a pattern different from the one that has been used historically can give rise to a new word, such as older replacing elder (where older follows the normal pattern of adjectival superlatives) and cows replacing kine (where cows fits the regular pattern of plural formation).