In Mexican Spanish,
- "Ocho días" = One week
- "Quince días" = Two weeks
- "Una quincena" = Two weeks (one fortnight)
- "Quincenal" = Every two weeks
Is this usage unique to Mexico?
(Some of the comments at this question made me wonder about this.)
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It appears ocho días = one week is common in some regions. I've never heard it in Argentina, nor the expressions mentioned in the other question (such as de aquí en ocho). It's either siete días or una semana.
The equivalence of 15 days to two weeks, on the other hand, is common here, and I gather it's common elsewhere. Quincena is mostly used in reference to wages (for someone who gets paid every two weeks); metonymically, it refers both to the span of time and to the money itself.
If you go to a doctor and s/he wants a follow-up in two weeks, s/he will say either «Te veo en dos semanas» or «Te veo en quince días». This is so common that I guess nobody would mistake quince días for fifteen actual days unless in very specific contexts.
Quincenal is also usually "once a fortnight", but it may imply "twice a month".
As Mexican I can answer the same way the given answer in the post
A week is not made from 8 days, but when something is bound to happen into a week from the current day, we will say "8 days from/starting today"
This means that "de hoy en 8" : "8 days from today" means that something is happening one week ahead.
Basically you count the days and add 1 (to include the current day in the counting)