How/when does one use "a most"?

"Most" in this context means "very." As in "a very wanted man." It functions as a "superlative."

This usage is considered "dated" and is not recommended for modern contexts. I am providing this explanation for people who want to read (or possibly write) earlier "period" novels or shows.

This is a special kind of absolute superlative. The OED gives as its sense 2 of most adv. the following:

As an intensive superlative qualifying adjs. and advs.: In the greatest possible degree.
_{For the ceremonial titles, Most Christian, Most Honourable, Most Noble, Most Reverend, etc., see those adjs.}

I don’t think it gets used all that very much anymore. One of the examples is:

1710 Hearne Collect. (O.H.S.) II. 351 ― A most vile, stinking Whigg.

Most can be used as 'very'... so 'a most' means 'a very'......

adv. Superlative of much; (intensifier): very. e.g. a most impressive piece of writing. (TFD/AHDEL)

Generally the definite Article 'the' is placed before any superlative. But there is an exception to this rule. This special use of 'superlative' is called 'absolute superlative' or 'superlative of eminence'.

Positive, comparative, and superlative degrees are used for comparison. But 'absolute superlative' or 'superlative of eminence', are used to express a very great degree of a quality, with no idea of comparison.

Consider:

1. A most beautiful sight...
2. A most interesting book...
3. He met me in a politest manner.

The question is misleading. Instead of asking for the meaning of "A most", it would be better to ask for the meaning of "most wanted":

A most wanted man

is to be read:

A [most wanted] man

and turns out to be:

A man who is most wanted