Linguistic structures are pairings of meaning and form. Any particular pairing of meaning and form is a Saussurean sign. For instance, the meaning "cat" is represented worldwide with a wide variety of different sound patterns (in spoken languages), movements of the hands and face (in signed languages), and written symbols (in written languages).
Linguists focusing on structure attempt to understand the rules regarding language use that native speakers know (not always consciously). All linguistic structures can be broken down into component parts that are combined according to (sub)conscious rules, over multiple levels of analysis. For instance, consider the structure of the word "tenth" on two different levels of analysis. On the level of internal word structure (known as morphology), the word "tenth" is made up of one linguistic form indicating a number and another form indicating ordinality. The rule governing the combination of these forms ensures that the ordinality marker "th" follows the number "ten." On the level of sound structure (known as phonology), structural analysis shows that the "n" sound in "tenth" is made differently from the "n" sound in "ten" spoken alone. Although most speakers of English are consciously aware of the rules governing internal structure of the word pieces of "tenth", they are less often aware of the rule governing its sound structure. Linguists focused on structure find and analyze rules such as these, which govern how native speakers use language.
Linguistics has many sub-fields concerned with particular aspects of linguistic structure. These sub-fields range from those focused primarily on form to those focused primarily on meaning. They also run the gamut of level of analysis of language, from individual sounds, to words, to phrases, up to discourse.
Sub-fields of structure-focused linguistics include:

• Phonetics, the study of the physical properties of speech (or signed) production and perception
• Phonology, the study of sounds (or signs) as discrete, abstract elements in the speaker's mind that distinguish meaning
• Morphology, the study of internal structures of words and how they can be modified
• Syntax, the study of how words combine to form grammatical sentences
• Semantics, the study of the meaning of words (lexical semantics) and fixed word combinations (phraseology), and how these combine to form the meanings of sentences
• Pragmatics, the study of how utterances are used in communicative acts, and the role played by context and nonlinguistic knowledge in the transmission of meaning
• Discourse analysis, the analysis of language use in texts (spoken, written, or signed)

Many linguists would agree that these divisions overlap considerably, and the independent significance of each of these areas is not universally acknowledged. Regardless of any particular linguist's position, each area has core concepts that foster significant scholarly inquiry and research.