A language is a dialect with an army and a navy

A dialect is a (usually: regional) variant of a language. The subject of study of linguistics is language as it occurs, and it so happens that language occurs in a continuous and always evolving range of varieties, often without clear borders. A nice example is German: one can travel from village to village, starting near Kortrijk or near Amsterdam, all the way to Szczecin, Vienna or Chur, and see only very small changes in the local dialect along the way, while over larger distances the dialects can no longer be mutually understood - actually, some are not known as dialects of German, but of Dutch, another well-established standard language within the same dialect continuum.

If a language is widely used, standard, 'official' forms will develop, with normative rules on spelling, pronunciation, grammar, and vocabulary. The term 'a language' is often used to refer to such a standardized form of language, mainly by non-linguists.

It is rarely by its linguistic properties that a particular language obtains recognition as 'a language'. Rather, it is a question of official backing.

So you shouldn't really ask this question to a linguist; ask a historian instead!

For example, the dialect continuum mentioned above covers the Netherlands, half of Belgium, Luxemburg, Germany, more than half of Switzerland, Austria, a region in Italy, and some small regions elsewhere; national interdialectal standards are developing in each of these states, but the two traditionally recognised languages are Dutch (in the Netherlands and most of the Belgian area) and German (elsewhere). They have both developed clear identities through centuries of teaching and usage. From a linguist's point of view however, they are just two variants that happen to have wider national use and recognition than any of the regional dialects.

This point of view is summarized in the well-known saying

  A shprakh iz a diyalekt mit an armey un a flot.

("A language is a dialect with an army and a navy.") This quote's earliest attested appearance, in Yiddish, is in an article by Max Weinreich (1945); earlier it appeared without the "and the navy" part. (Note that Switzerland and Austria do not border on the sea - they do not have navies.)

See http://www.edu-cyberpg.com/Linguistics/armynavy.html for details.

The example of German given above is very very good; "German" has incredibly wide variations within it. Swiss German movies or television, when played in Germany or Austria, require subtitles. A person with a working knowledge of both German and English will understand Dutch without very much difficulty. Dutch and Flemish (spoken in Belgium) are essentially the same language with two different names.

Head to Luxembourg and you find Luxembourgish, a hybrid of German and French.

In the Scandinavian department, the "Norwegian" language actually consists of two variants, Bokmål and Nynorsk, Bokmål actually being more similar to Danish than to Nynorsk.

Some dialects of English spoken in Scotland or Ireland are not understood without great difficulty by Americans.

In eastern Slovakia is spoken a variant of Slovak so influenced by the nearby Ukrainian that it is no longer mutually intelligible with Slovak, nor with Ukrainian. Meanwhile, Czech and Slovak are nearly mutually intelligible.

Speak Portuguese slowly to a Spanish-speaker and they'll understand, sorta. Same with Catalan or Italian.

And Chinese? Ha! No one even tries to claim that the so-called "Dialects" (Mandarin, Cantonese, Wu, &c.) are at all the same language.

And what about the little dialects that claim they actually are languages? Bayerisch, Occitan, Galego, Provençal, Friulian... These are languages without armies.

Trying to draw lines between languages is like trying to chop up the rainbow. Blue and Green are distinct colors, but what about Turquoise? Is it blue? Is it green? Is it its own color? To try to argue that is nothing but pointless.