Saudade is a Portuguese word that appears here and there in English writings. Usually the writer will give brief explanation, something along the lines of "The Portuguese word for 'the presence of absence'" (-Spider Robinson). It is often be touted as 'untranslatable'. The result is that saudade is seen as a type of bittersweet super-nostalgia, bigger and better than anything that the English speaking world can truly understand. How poetic!
Well, I have no poetry in my soul, but I do have a Portuguese dictionary1. And it turns out that the definition of saudade translates very neatly into English.
Saudade (do ant. soedade, soidade, suidade - Lat. solitate, com influência de saudar), s.f. lembrança triste e suave de pessoas ou coisas distantes ou extintas, acompanhada do desejo de as tornar a ver ou a possuir; pesar pela ausência de alguém que nos é querido; nostalgia; (Bot.) nome de várias plantas dipsacáceas e das respectivas flores; (no pl.) lembranças afectuosas apessoas ausentes; (no pl.) cumprimentos.
And here's the translation. No translation is 100 percent indisputable, but in this case, it's a pretty straightforward and bland task.
Saudade feminine noun. A memory sad, but sweet, of persons or things that are distant or lost, accompanied by the desire to see or have them again; To feel grief over the absence of a person whom you love; nostalgia; (Bot.) The common name of various plants of the family Dipsacaceae (the teasels), and their flowers; (in the plural; saudades) Affectionate remembrances of those who have died; (in the plural; saudades) may be used as a greeting2.
Well, that sucked the romance right out, didn't it? But as you might expect, the themes of missing those you love and homesickness are a major component of poetry and songs, including the famous Portuguese fado. When English speakers have tried to translate these songs they came upon the problem that the writer/singer is filling the piece with strong emotion and, hopefully, beauty. In English the phrases "I have longing" and "I am with longing" sound awkward and wrong, with a distinct lack of beauty. If you simply replaced saudade with the word 'nostalgia' you would usually have a good translation, but it would sound clumsy. Faced with the difficulty of translating a word that just doesn't fit grammatically into English, the English-speaking world decided to give up. Just use the Portuguese word that fills the grammatical position so well, and pretend that we couldn't translate it if we tried.
Longing, nostalgia, and missing someone who is not there are all well developed ideas in English, and there's no need to pretend that saudade is anything that we don't speak about in our everyday lives. We tend to emphasize the feeling as a verb (I long for you, I miss you), and avoid using it as a noun (I have longing, I feel nostalgia for you). But the idea is the same.
Having said all that, there is the idea of Saudosismo, an artistic and philosophical idea that appears in the early 1900s. This is a highly poignant nostalgia for the way things used to be. The longing for the old folkways, the idealization of the life that once was, the desire to escape the modern world and live as the noble Portuguese once did.
This movement has added an extra dimension to the word saudade. Sometimes it is used in the context of glorifying lost ways of life, reaffirming the Portuguese cultural identity, and bringing the nostalgia for Portuguese history to an almost religious plane. But this is not a sense that English speakers really have any reason to use. The English mythology of saudade doesn't really have any Portuguese counterpart.
1. No, not a Portuguese-English Dictionary. A Portuguese Dictionary. The Dicionario Universal Lingua Portuguesa, published by Cabo Verde Editoria. In the spirit of full disclosure, this dictionary was published in Cape Verde, and is not the 'most official' dictionary of the Portuguese language. It does have the benefit of being a Universal Dictionary of the Portuguese language, which means that it is 1. Very Big and 2. includes both Portuguese and Brazilian words and alternate meanings of words.
2. I have never heard Saudade used as a greeting in Cape Verde, but apparently it's not too uncommon in Portugal. Rancid_Pickle reports knowing a young man who used this greeting exclusively, and graceness recounts having heard it used as a greeting among those attending a funeral. That seems quite appropriate.