In really general terms, one can look at language classification as akin to biological taxonomy.
Dialects are like varieties and subspecies, which are grouped together to form languages. The distinction between dialect and language is slightly nebulous, but includes mutual intelligibility and lack of separate national history and literature. E.g., Australian English is a dialect of English because it is mutually intelligible and quite obviously derived from British English. Dutch and Flemish, mutually intelligible, have different national standards and a long history of separate existence with different literature, and so are usually considered separate languages.
Languages are grouped into families where close connections of vocabulary, grammar, and syntax are obvious. For example, English, Dutch, Flemish, and German are in the West Germanic subfamily of the Germanic family (as opposed to the North Germanic tongues like Danish and Swedish).
Families are in turn grouped into phyla (singular, phylum) on the basis of more distant relationships. The Germanic languages are grouped with the Romance languages (French, Spanish, Italian, etc.), the Slavic languages, the Celtic languages (Irish, Welsh, etc.), the Iranian languages (Persian, Tadzhik, etc.), and the Indic languages constituting most of the languages spoken in India, Bangladesh, and Pakistan as the Indo-European Phylum.
The term “language stock” is sometimes used, with varying meanings depending on the preferences of the anthropologist or comparative linguist using the term.
Some of the major language phyla are:
[ul][li]Indo-European – A large group of major world languages, spoken in Europe and South Asia. and having spread from there to a lot of places, including the Americas and Australia[/li][li]Sino-Tibetan – Chinese languages and a lot of smaller East Asian languages which people without a special interest in the area are not familiar with[/li][li]Afro-Asiatic, formerly called Hamito-Semitic – Arabic, Hebrew, most of the now-extinct languages of the Ancient Middle East, and a large group of languages throughout the northern half of Africa, including Tuareg, Amharic, and Somali.[/li][li]Finno-Ugric and Altaic, which may or may not be joined as Ural-Altaic are two other groups of some importance. The first includes the Finnic family – Finnish, Estonian, and smaller relatives – and a group comprising Hungarian and two West Siberian languages. The second includes the Turkic languages of Turkey and the steppe republics, along with Uighur (the language of Sinkiang); the Mongol languages; and the Tungus languages such as Yakut and Manchu.[/li][li]Niger-Congo – Includes all the Bantu languages of Africa from Gabon and Kenya south to the Cape, and several smaller families across West Africa. Ibo, Fulani, Mandingo, and Ewe are examples of the latter.[/li][li]Malayo-Polynesian – A very broad though thinly stretched group, it includes Malagasy, Malay, Indonesian, Maori, the Filipino languages such as Cebuano and Tagalog, and most of the languages of the Pacific islands[/li][li]Dravidian – Geographically small but important, the Dravidian languages are spoken in South India and Ceylon, with a few outliers in northern India and Pakistan.[/li][li]Uto-Aztecan – Native langauges spoken from the American Southwest through most of Mexico[/li][li]Athabascan – Native languages of much of Western Canada with groups spread out from there[/li][li]Macro-Siouxan** – Includes the Iroquois and Sioux languages of America, with a quite widespread original territory[/li][li]Andean-Equatorial** – A lot of the major Indian languages of South America are in this group: Guarani, the co-official language of Paraguay; Quechua, the language of the Inca Empire; Aymara, common in Bolivia and southernmost Peru; Tupi, something of a lingua franca among the Indians of much of Brazil. Arawak also belonged to this group.[/ul][/li]
Many languages are members of much smaller phyla or language isolates with no clear relatives. Georgian (in the Caucusus, not around Atlanta) is a member of the small Kartvelian phylum; Basque (famously), Korean, and Japanese are examples of isolates.