Morphology is a field of linguistics focused on the study of the forms and formation of words in a language. A morpheme is the smallest indivisible unit of a language that retains meaning. The rules of morphology within a language tend to be relatively regular, so that if one sees the noun morphemes for the first time, for example, one can deduce that it is likely related to the word morpheme.
There are three main types of languages when it comes to morphology: two of these are polysynthetic, meaning that words are made up of connected morphemes. One type of polysynthetic language is a fusional or inflected language, in which morphemes are squeezed together and often changed dramatically in the process. English is a good example of a fusional language. The other type of polysynthetic language is an agglutinative language, in which morphemes are connected but remain more or less unchanged – many Native American languages, as well as Swahili, Japanese, German and Hungarian, demonstrate this. At the other end of the spectrum are the analytic or isolating languages, in which a great majority of morphemes remain independent words – Mandarin is the best example of this.
This can be a confusing concept, so an example may be helpful. Looking at the morphology of English, which is not a particularly inflected language in its modern form, but retains a number of remnants, we could create the word frighteningly, which is made up of four morphemes: fright, which is a noun; en, which converts the noun to a verb; ing, which converts it to an adjective; and ly, which converts it to an adverb. Over time, languages tend to become less and less inflected – particularly when a lot of intercultural contact occurs. In morphology, this is because the languages become creolized as various pidgins used for communicating between disparate groups become natively spoken, and inter-communication in the pidgins is facilitated by dropping inflections.
Although you may be used to seeing certain forms in a specific context – such as conjugations at the end of a word – they can express themselves in a number of different ways. Aside from the English use of prefix and suffix, words can also be inflected by changing the sound of a vowel – called an umlaut – or by placing an affix right in the middle of the word. Affixes can also be quite lengthy, not just little bites of sound – in Quechua, for example, there are a number of two-syllable affixes. Though most people never formally study morphology, it is something native speakers understand intuitively. Any time a person learns a new word and immediately comes up with any number of forms for that word – past tense, plural, a noun form – they are applying the rules of morphology subconsciously to determine what the new form should be.