Periódico vs. diario

The words "periódico" and "diario" are Spanish terms that both refer to what English speakers regard as a 'newspaper'. However, they have subtly different contexts and uses in the Spanish language.

Periódico

This is the general term for 'newspaper' in Spanish. It could be published on various frequencies, not necessarily daily.
¿Has leído el periódico hoy?
(Have you read the newspaper today?)
Compra un periódico en el quiosco.
(Buy a newspaper at the kiosk.)
El último número del periódico incluye una interesante entrevista.
(The latest issue of the newspaper includes an interesting interview.)

Diario

'Diario' refers specifically to a daily newspaper or journal. It's derived from 'día', meaning 'day', implying its daily issuance.
Leo el diario cada mañana.
(I read the daily every morning.)
El diario local informó sobre el evento.
(The local daily reported on the event.)
"El País" es un diario español muy conocido.
('El País' is a well-known Spanish daily.)

Summary

'Periódico' and 'diario' both mean 'newspaper', but whilst 'periódico' is a generic term used for any newspaper regardless of its publication frequency, 'diario' specifically refers to a daily paper. Therefore, all 'diarios' could be called 'periódicos', but not all 'periódicos' are necessarily 'diarios'.