There are two main ways that words can be built up of many morphemes. Agglutinative languages build words by "gluing" morphemes together essentially unchanged. Fusional languages build words by "squishing" morphemes together, often changing the morphemes in the process. European languages tend to be fusional languages, while Native American languages tend to be highly agglutinative.
A synthetic language is one that has more than one morpheme per word, and that covers most languages. A polysynthetic one has a more extreme degree of morpheme joining, and is often taken to mean that there is some degree of incorporation, such as of noun and verb together in the same word.
Examples of polysynthetic languages include Inuktitut, Mohawk, and numerous other languages of North America and Siberia. Languages with a high degree of synthesis but without being incorporating include Basque and the Bantu languages. According to some linguists French can be classed as highly synthetic: a phrase like je ne le sais pas is all one word because the preceding clitics are actually word inflections. It is structurally similar to a single Bantu word. If this is true, French is thus far and away the most synthetic Indo-European language.
The terms synthetic and polysynthetic in this sense were first used by Edward Sapir in the 1920s.