How it all started
The beginning of the serious classification of languages pretty much began with Indo-European, although even before that, there had been work on other language groups.
How was Indo-European discovered? Sir William Jones, a British judge in India, compared lists of the same words in Arabic, Hebrew, Sanskrit (an ancient language of India and Hinduism), Avestan (old Persian, anscestor to the language now spoken in Iran), Classical Greek, Latin, Gothic (the language of the Germanic tribes who sacked Rome), Old Irish, and Turkish. Jones didn't nothing more than compare words in the languages, but through this, discovered that, except for Arabic and Hebrew (which are related to each other), and Turkish, all the languages were related to each other.
Indo-European today is very well-accepted--virtually no one doubts its existance--yet the methods which found this family are the very methods that today are criticized. This happened because of what happened with the study of Indo-European in the 19th century. After Indo-European was discovered, many people started studying the languages, and began finding regularities in the differences between words in the languages. Later still, they started reconstructing Proto-Indo-European, the language that gave rise to all the Indo-European languages. Since Indo-European had already been discovered, no one needed to classify the languages, and they began to think that classifying languages involved reconstructing them, while reconstructing is really something that should be done after languages have been compared and classified.
That's a brief summary of the controversy. For more detail, and an excellent introduction to historical linguistics and language classification, I highly recomend "The Origin of Language: Tracing the Evolution of the Mother Tongue" by Merritt Ruhlen.
Now, the genetic classification of the world's languages.
Back to Languages and Linguistics.