was the dominant approach in linguistics
for the earlier part of the twentieth century. It was overtaken by Noam Chomsky
approach from the 1950s on.
Traditional grammar taught that there was good and bad, or correct and incorrect, language. Latin was held up as the paragon of rational grammar, and a common object of study was to read and write Latin well, or to understand another language within the terms of Latin. In the nineteenth century a historical approach took over, as linguists came to unravel the history of the Indo-European language family. Etymology and the genetic relationship between languages became the main area of study.
Ferdinand de Saussure, a linguist whose contribution to the Indo-European field included the important postulate of laryngeals (confirmed years after his death), wanted to distinguish how people actually used language from the historical question of how it had come to be used that way. This is the key distinction of the synchronic versus the diachronic. Synchronically, we might use a word like hysterical without being aware of its origin, and when we say someone is hysterical we're probably not claiming anything about their womb. Diachronic information is usually only available to specialists or in dictionaries, whereas the synchronic aspect of language is what's present to us as common speakers.
Langue and parole are usually left untranslated, as it would cause too much trouble to explain if we used heavily loaded terms like 'language' and 'speech' and tried to distinguish them consistently. Langue is how we do, in theory, know how to speak; parole is what actually comes out on a particular occasion: with a slip of the tongue, speech impediment, hesitation, false start, or whatever. In Chomsky's terms, these are similar to what are called competence and performance. But Saussure's langue is a social commonality, while Chomsky's competence is an I-language in an individual's brain.
The distinction between paradigm and syntagm is that a syntagm is an extended structural description, while a paradigm is a drop-down list of specific things that can go in each slot of the description. For example, The linguist saw the pelican has essentially the same syntagmatic structure as A møøse bit my sister, but different choices at each point. The paradigm of the first and fourth places is that of a determiner: a word from the closed set a, the, this, that, my, your, ..., whereas at the third slot there is a verb, a member of an open-ended class of words. The verb has its own paradigm of how its past tense is formed. All past tense verbs are syntagmatically V + Past, but there are different choices of realization of Past (-ed, -t, vowel change, a mixture) at that point.
Other aspects of Saussure's formulation given above are more semiotic, that is part of a general description of how signs refer to things, and he was building on the work of the philosopher Charles Sanders Peirce. They are less specific to linguistics.
In America structuralism was taken up by anthropologists like Franz Boas and Edward Sapir who needed to analyse the very un-European structure of native American languages, and found that a structuralist approach helped them understand things like kinship systems. Other influential American linguists like Leonard Bloomfield, Zellig Harris, Kenneth Pike, and Joseph Greenberg found the structuralist approach extremely fruitful.
In Europe the main structuralist linguists included Roman Jakobson and Prince Nikolai Trubetzkoy, leaders of what is known as the Prague School. They disagreed with Saussure's complete severance of the synchronic and diachronic, and especially applied structuralist ideas to phonetics, showing how sound changes across time could be motivated by changes in structural paradigm. For instance, in Old English the sound V only occurred between vowels, and the F sound never did: both were written F because they couldn't contrast, they were the same phoneme. With the Norman Conquest, French words came in that did have this contrast, so the phonology of Middle English was restructured to treat F and V as separate sounds.
Chomsky's theories could be described as structuralist, but he was concerned with two interacting levels, deep structure and surface structure, and he attacked the older structuralist tradition as taxonomic, that is collecting information about languages without really trying to understand the workings. So, following him, the term 'structural linguistics' or 'structuralism' is usually confined to the other schools and methods mentioned above.