Morphology is a subfield of linguistics that deals with the form and internal constituent structure of words of a given language. Since words are formed of morphemes, we can also say that Morphology is the study of morphemes. We can define a morpheme as the minimal unit of form (structure) and function (meaning) in a language. Note that there are problems with this so-called standard definition of a morpheme, as we have cases of null morphemes (function without a form) and empty morphemes (form without a function) in many languages. Besides this, morphemes might not always have a clear, one-to-one matching of form with function (ablauts, portmanteau morphemes and allomorphs).
Such an analysis of morphemes of a language can serve many purposes. We get to understand how words are formed (inflected and derived) in a language. A study of borrowed roots and their origins is helpful in diachronic studies. When we do a typological classification of languages of the world, we can use the morphemic structure of languages to classify them as synthetic, polysynthetic, agglutinative, isolating, etc. Such a classification is useful because languages that fall under each category have striking similarities on other linguistic levels as well, and this can potentially uncover other kinds of relationships between languages. And if we are able to establish a strong relationship between seemingly unrelated languages, we can get a better picture of Universal Grammar (language faculty).
Language is an ever-changing, dynamic system. Changes in language also happen at the lexical level. Neologisms are fairly common in almost all languages. But new lexical items that enter the system have to, inevitably, follow the morphological structure of the given language. For instance, the position of grammatical morphemes is fixed in a language and has a correlation with the word order, which in turn has a relationship with the prosody of a language. The native speakers of a language have intuitions about these things. As linguists, however, one of our major goals is to describe this innate knowledge.
Morphological studies also play a role in syntactic analysis or understanding the grammatical structure of a language (morpho-syntax) as well as in the study of phonology or sound system of the language (morpho-phonemics).