An analytic language is a language that conveys grammatical relationships without using inflectional affixes. A grammatical construction can similarly be called analytic if it uses unbound morphemes, which are separate words, and/or word order.
A related concept is the isolating language, which is about a low number of morphemes per word, taking into account derivational morphemes as well. A purely isolating language would be analytic by necessity, lacking inflectional morphemes by definition. However, the reverse is not necessarily true: a language can have derivational morphemes while lacking on inflectional morphemes. For example, Mandarin Chinese has many compound words, giving it a moderately high ratio of morphemes per word, yet since it has almost no inflectional affixes at all to convey grammatical relationships it is a very analytic language.
The term "analytic" is commonly used in a relative rather than an absolute sense. English has lost much of the inflectional morphology of Proto-Indo-European over the centuries and hasn't gained any new inflectional morphemes in the meantime, which makes it more analytic than most Indo-European languages. E.g., while Proto-Indo-European had inflections for eight cases in its nouns, English lost most of them, conserving only the genitive (possessive) -'s (and even this has been recently reanalized as a clitic, separate word: e.g. "Wright and Hermann's hypothesis").
For comparison, nouns in Russian inflect for at least six cases, most of them descended from Proto-Indo-European's cases, whose functions English translates using other strategies like prepositions, verbal voice and word order instead.
However, English is also not totally analytic in its nouns as it does use inflections for number, e.g. "one day, three days; one boy, four boys". Mandarin Chinese has, in contrast, no inflections in its nouns at all: compare 一日 yī rì 'one day', 三日 sān rì 'three days' (literally "three sun"); 一个男孩 yī ge nánhái 'one boy' (lit. "one [entity of] male child"), 四个男孩 sì ge nánhái 'four boys' (lit. "four [entity of] male child").
See also 
- Auxiliary verb
- Free morpheme
- Isolating language
- Zero-marking language
- Synthetic language
- Linguistic typology
- Li, Charles and Thompson, Sandra A., Mandarin Chinese: A Functional Reference Grammar, University of California Press, 1981, p. 46.