In the field of linguistics, an archaism is an archaic word, morpheme, or phrase that still appears in some contexts, and is generally intelligible to a fluent speaker.

This happens a lot. An obvious example is let's go, a very archaic form of saying something like 'we're going'. We would (almost) never say 'let us go', but the contracted form is tradition, and it has no appropriately polite and concise replacement, so we just go with what has worked for centuries.

We see archaisms frequently in law (e.g., heretofore, thereupon, and whereabouts) and religion (verily thou shalt be fed, to have and to hold), in attempts to sound more formal, in famous writings from days past, and to give proper tone to modern historical narratives.

It should be noted that archaisms often have a tone of their own; in Robin Hood Daffy (1958) Porky Pig asks Daffy Duck "Prithee, O traveling clown, couldst thou directest me to Robin Hood's hideout?" This has no serious resemblance to the English spoken in the time of Robin Hood, and couldn't, as modern audiences don't speak early modern English; however, we know that 'prithee', 'O', -st, and 'thou' are markers of 'old timey speech', so it works. Likewise, when reading a modern work mimicking Victorian speech, 1930s Chicago slang, or any other now-defunct speech pattern, there are recognized markers that we use, and any number of perfectly normal (for the time) words and phrases that we would never think to use.

Some works are so famous that we continue to read them, and hopefully comprehend them, for centuries past their publication. Reading the King James Bible or the works of Shakespeare will take some work, but not as much as it might, as many of their otherwise archaic language forms are still in the popular conscious today; we might never say anything at all like

Two households, both alike in dignity,
In fair Verona, where we lay our scene,
From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean.

but we clearly recognize it, and probably understand it. In fact, the archaic words are often not a problem in comprehension; "take it in what sense thou wilt" is perfectly intelligible, whereas "where civil blood makes civil hands unclean" might cause a double-take despite a comparative lack of archaisms.

Archaisms often appear in idioms (e.g., hoist by one's own petard), and they likewise include fossil words, cranberry morphemes, and orphan negatives. Eventually many words do die out completely, or so nearly so as to make no difference; at this point they are no longer archaic, but obsolete.