What is logos?

The basic meaning of the noun logos is related to the verb legein, "to gather", "to lay down", "to pick". In this sense, logos simply means a "gathering", a "collection". Hence, the English word "anthology" does not mean "science of flowers" nor "a discourse on flowers" (anthos meaning "flower"), but a bundle, a gathering or a collection of poems. (Cf. "terminology", "mythology", "ideology", "chronology", which all are collections, and not sciences.)
The more derivative, later meanings of logos relate to this basic sense more or less loosely. So logos means speech and discourse, probably in the sense of something gathered and organized. (Despite the -logy suffix in modern languages, and despite the names of some particular sciences, "science" as a word on its own is rendered with by epistêmê and not by logos.) In a connected way, logos can mean definition and account, the latter both in the sense of the rational explanation or reason of something (cf. logon didonai), and in the mathematical sense of reckoning. Hence the sense proportion as in "the ratio of 2 to 3". (See the Latin ratio in the phrase "irrational number"; Cf. the Arabic "(jaḏr) 'aṣamm" ("deaf (root)".)
Further, as James Boll pointed out, logos came to mean the capacity which enables one to produce such a speech or account: the faculty of reason. The associations with Divine Reason, Providence or Incarnation, are vaguely found in some Presocratics (especially Heraclitus), but more clearly in Stoic and indeed Christian doctrines. (Cf. en arkhê ê ho Logos, In principio erat Verbum, etc., John, 1:1) This association of logos with Divinity is totally absent from Aristotle's use. (By the way, this was part of the conclusion of my Ph.D. dissertation.) Before that logos does not even mean "word", but "sentence".
Finally, to me, the most unexpected meaning of logos is esteem, or consideration. But again, see the English phrase "to take into account", which does not only mean "to include in a calculation", but can also mean "to take seriously", "to pay attention".)