I'm taking a typology class this semester, so I thought I would post on the four-way morphological typology typically employed when discussing languages. Typology is concerned with the limited number of patterns that languages use, and how the use of these patterns (especially which are most common) tells us about things which are universal in language. One way of categorizing languages is according to morphology (how languages put together words). Two indeces are typically used: the index of synthesis, which refers to how many morphemes are contained in a typical word, and the index of fusion, which refers to how segmentable morphemes are and how transparent the morphophonological changes are.
On the index of synthesis, we have two poles: isolating and polysynthetic. An isolating language typically has one morpheme per word (i.e., there is a separate word for every grammatical function, e.g., Chinese or Vietnamese). A polysynthetic language typically has many morphemes per word, and entire sentences/complete thoughts are a single word (e.g., Blackfoot). As an example, the Blackfoot word kitakitamatsinopoao(a), which is used as "goodbye", literally translates as "You (pl.) and I will see each other again". Sometimes this is classification broken down further, either into synthetic (1-3 morphemes per word) vs. polysynthetic (4 or more morphemes per word), or into synthetic/polysynthetic (many morphemes, but only one lexical root) vs. incorporating (words have multiple lexical roots, e.g., Chukchi).
The index of fusion also has two poles: agglutinative and fusional (or inflectional). Agglutinative languages have many morphemes in a word, but each morpheme contributes only one grammatical meaning, and each morpheme is clearly segmented, e.g., Turkish. English, when it uses multiple morphemes in a word, is usually agglutinative. "Wonderfully" is easily segmented into wonder-ful-ly, and each morpheme contributes a single meaning. Fusional languages, on the other hand, tend to use fewer morphemes per word because each morpheme contributes multiple grammatical meanings, e.g., Russian or Spanish. In Spanish, the -o in "hablo" contributes the meanings "1st person", "singular", "present", and "indicative mood". It's a single sound, so it's not possible to segment it at all; it simply has all those meanings rolled into one sound.
Now, of course there are essentially no language that fit neatly into one category or another (including the languages I cited as example in each category), which is why we organize the four traits into sliding scales rather than leaving them as strict categories. Some languages are more analytic, some or more synthetic. Some languages are more agglutinative, while some are more fusional.