Nostalgias e morriñas
It is an oft-repeated myth that the Portuguese and Galician word saudade “cannot be translated” into other languages. It certainly can! It just may take more than one word to do so adequately.
The basic sentiment is one that mixes together aspects of melancholy, loneliness, love, distance temporal or spatial, and permanent loss. Even though few languages (but not quite none: see below) have just a single word encompassing all of those, it is no mean feat to combine multiple words to produce just the right matiz, no matter the language.
Because of the difficulties in finding precisely the right shade of meaning in translating the Portuguese and Galician word saudade into Spanish, the Spanish have appropriated the word for themselves: an entry for saudade can be found in the DRAE, where it lists as senses the words soledad, nostalgia, añoranza. The last of those three words, añoranza, is perhaps best translated into English as “missing”, “longing”, or especially “pining”. Saudade has elements of all those and more, which is why the Spanish often enough just give up and use saudade directly. :)
Certainly nostalgia alone isn’t good enough, since saudade is the melancholy feeling of loneliness that comes from missing someone or something or some place or some bygone era, combined the critical connotation that that thing which is missed was quite dear to you yet now is gone forever and shall never return.
You need a word or phrase that mixes together all three of love and loss and distance. Nostalgia alone cannot cover all that territory, nor can the English refrain the dear departed, which are people not feelings.
The exact etymology leading from some Latin word like solitate to modern saudade is disputed1, 2 but the presence of intermediate forms in literature like soëdade, soidade and suidade suggest a word in flux. It is reasonably simple to hypothesize the lenition of the Latin l from a word of Latin origin (as occurred with the loss of l in the definite articles and dative pronouns derived from Latin ille/illa/illum), but again, there may be more factors involved (like associations with words like suave).
It is not even certain that saudade has its origin in Portugal; it may have originally come from Galicia, and is certainly associated with that land. From Wikipedia:
Para Joaquim de Carvalho, a explicación desta localización non é allea á orixe céltica. Unamuno supón a orixe da saudade nas formas da paisaxe galega, "un paisaje habitable, que seduce como un nido incubador de morriñas y saudades". Na mesma liña, Gerald Brenan ve na saudade unha orixe indubidablemente climática: os ventos atlánticos proporciónanlle a Galicia o mesmo espírito lánguido que a Irlanda e ás Illas Hébridas. Isto atópase nos diálogos de Martín Codax co seu mar.
Although there are other languages with words that are more or less close to that of saudade, apparently the language with a word that is closest is Welsh, which per Wikipedia’s article on the word has hiraeth:
Existe o mito de que a palavra 'saudade' só existe na língua portuguesa e como tal não pode ser traduzida correctamente. De fato, o sentimento de saudade não tem significação específica em outras línguas.
[...]
A palavra que mais se assemelha é "Hiraeth" do galês e é a única com uma conotação textualmente semelhante.
Saudade is not usually a level of sadness and anguish strong enough to quite qualify as actual depression. However, there is a word which is. From Wikipedia again:
Saudade is also associated with Galicia, where it is used similarly to the word morriña (longingness). Yet, morriña often implies a deeper stage of saudade, a "saudade so strong it can even kill," as the Galician saying goes. Morriña was a term often used by emigrant Galicians when talking about the Galician motherland they left behind. Although saudade is also a Galician word, the meaning of longing for something that might return is generally associated with morriña.
And from the Galician Wikipedia page:
A frecuente identificación entre saudade e morriña nace da confusión de termos cercanos, pero o que caracteriza á morriña é a tristura depresiva, mentres que a saudade está caracterizada pola carencia de significación psicolóxica. A identificación da saudade coa Sehnsucht dos alemáns, proposta no seu día por José Luis Varela, afástaa da súa acepción de nostalxia dun ben perdido para considerala a procura dun obxecto descoñecido que se sente necesario. Celestino Fernández de la Vega, pola súa parte, identifica a saudade coa angustia como a entendía Heidegger, sen reparar en que a saudade carece da dimensión temporal da angustia.