The loss of a language is like the loss of a cherished museum or library: a language bears, in its lexical and semantic features, in its ways of saying things, a significant measure of the civilization of its speakers. The loss of Damin, for example, the initiates' language of the Lardil people of Morninton Island, North Queensland, amounts to the loss of a tradition of semantic relations comparable to that embodied in the very best thesaurus, or in the entire output of the anthropological tradition of componential analysis of the fifties and sixties.
While languages are being lost at an alarming rate, this is not a time for pessimism. Many local language communities are mobilizing to maintain and foster their linguistic traditions, and there are successes, including among many others the immersion programs of Maori, Hawaiian, and Mohawk, the Irish of Belfast, and the language reclamation project of the Miami-Illinois people of Indiana.
I like to focus on people, and the effect the loss of their language can have on them. Indigenous persons are often viewed as museum entities, as things objectified, that don't really coexist with us in our Western world. But they do. The work I do in Nicaragua focuses on the empowerment of the people(s).
You may hear the argument that it is actually more "useful" for a given minority language group to just forget about its language and use the dominant language in order to have better access to "success." You often see this argument in discussions about African American English. In the end, this discussion is about assimilation: "if you don't assimilate, you don't survive" (mostly in economic terms). But this is a total fallacy. It is true that some individuals succeed by assimilating—that is, by associating with the ways and language of the dominant group—but the majority, the group, the people, won't. And these people will still be marginalized (will not "succeed")—AND will have lost one of their treasures, one of their principal sources of identity and pride: their language.
All languages can express abstract ideas, science, and art equally, though maybe differently. No language is better suited than any other to be used for science; technical vocabulary has been introduced in every language. Every language had to adapt to new realities. But that doesn't mean that a given language is less apt to be used for "success."
When a language is lost, people and peoples are deprived of one of the major assets of cultural identity. It is part of what is often referred to in Latin America as "acculturation": in other words, cultural genocide. Among the basic rights of humans is the right to their own culture, and to their own language. Let's defend that right.
There is still a great deal of debate about whether culture and language are separable, and, even deeper, whether language forces you to think in particular ways. You can talk with many people—the Irish come to mind, for example—who have adopted English as their language but still consider themselves to be part of their heritage culture. So it is clearly possible to have the culture survive without the language. But language is clearly the most efficient carrier of a culture—it is virtually impossible to speak the language and not carry on the culture. This is not to say that cultures don't change—there is nothing odd about a Native American using a camcorder to tape a dance competition. In this country only the Amish believe that technology should not change.
Speakers of the majority language tend to ignore it until some amusing miscommunication occurs. The most reasonable scenario for the survival of endangered languages is to have bilingual communities. Usually the majority language (or even the global language of English) is necessary for day-to-day living. But most communities throughout human history have tended to be bilingual; it just needs to be seen as the norm again.
One word of caution: it is an extreme misconception to think that languages "amalgamate" or in any sense merge; under any ordinary circumstances they do not. Sure, all languages borrow individual words from one another, but words are relatively superficial; it's the structure of sentences and of words (verb forms, for instance) that are the real "bones" of a language, and they aren't usually affected by mere contact with other languages. (Even English, which is supposed to be very "mixed," is actually an unremarkable West Germanic language; all the "grammar"—literally all of it—is Anglo-Saxon, though much of the highfalutin' vocabulary has been borrowed in from elsewhere.) What makes language death interesting, and upsetting, to us linguists is that language death is precisely the kind of abnormal situation in which the basic structure of a language does get eroded.
Languages begin to disappear as a consequence of individual family decisions. Sometimes those decisions are made not because the community language is without value, but rather with the idea that some other language has a positive value and should be acquired by the children. What many parents fail to understand is that children will acquire the language(s) they hear, so if the community language is not spoken at home, they may not learn it. There is often an unconscious belief that children will "naturally" acquire the community language (it is, after all, the language that defines the community and that everyone speaks) but that they may need special help to learn a non-community language. So parents and older siblings may speak to children in the language they need special help with—neglecting to provide a sufficient context for acquiring the "home" language. By the time they realize what's going on, it may be too late to change the situation.
Another belief that often leads to language loss is that it is difficult to be bilingual, especially for children, so only one language should be used with them. This belief frequently is in direct contradiction to the experience of the adults, who may in fact be perfectly bilingual in both the home language and a second language of wider communication. However, their own experiences while acquiring the second language may have been painful (they may have been subjected to ridicule in school, for instance), and they quite naturally wish to avoid this kind of situation for their children. This, coupled with the belief that the children will naturally acquire the home language, may lead to even less use of the "home" language at home.
Language loss can be reversed, but that is a consequence of individual, usually conscious, family decisions. When parents and other caregivers decide that children should learn both the home language and a second language, speak to them in both languages, and insist that they answer in both languages (sometimes at slightly different ages, to avoid confusion), children can successfully become bilingual and language loss can be reversed. Coming to a realization that some children (but not your own) must learn the home language for it to survive is not quite enough—your own children are the keys to language survival.
As the other respondents have so ably said, language loss is the loss of unique grammars, meanings, great oral literatures, cultural knowledge, and even identity and self-respect. I'd add to that the loss of communication practices and, as a language dies, the thinning of bonds between the generations. Talk to families in communities whose languages are dying and you will hear the anger of the old at the young with whom they cannot communicate fully; the fear of semi-speakers to use their language because they may be laughed at for a mistake; and the self-hate of those who never had a chance to learn their language at all.
Like Elena, I believe that the best arguments for supporting minority languages can be found in the hearts of the people whose languages they are. As a linguist, I am surrounded by people who love languages, but I have never seen a depth of passion for a language as intense as the passion that indigenous language activists feel for their own ancestral tongue. At gatherings of indigenous people who are working on language revitalization, you can feel the grief of people talking about their lost or dying language, and you can observe the palpable joy of those who are learning to speak that language again. Language revitalization is one of the hardest and longest tasks there is in the world, and I often get discouraged about the chances that language loss can be turned around in this modern world where English, especially, smothers all languages around it. But all I have to do to recover my optimism is get together with the community people who are working to save their languages. The strength of their dedication, and their pleasure in each achievement, small or large, brings back my own confidence that it can be done, and also that the effort itself brings its own rewards. My favorite saying in this regard is by L. Frank Manriquez. She and several others are trying to reclaim their ancestral language, Tongva, once spoken in the Los Angeles area, and now surviving only in a few dusty books and field notes. Coming away once from a language conference of indigenous-language activists, she said happily, "How can it be hopeless when there is so much hope?"