The opposite of a synthetic language is an analytic, or isolating language. Synthetic languages which are not agglutinative are called fusional languages; they combine morphemes by "squeezing" them together, often changing the morphemes drastically in the process.
"Agglutinative" is sometimes used as a synonym for synthetic, although it technically is not. When used in this way, the word embraces fusional languages and inflected languages in general. It is also worth noting that the distinction between an agglutinative and a fusional language is often not a sharp one. Rather one should think of these as two ends of a continuum, with various languages falling more toward one end or the other.
As you can see, agglutinative languages are not entirely grouped by the family (although Finnish and Hungarian are related, as are possibly Japanese and Korean). Rather, convergent evolution had many separate languages develop this property.