Let's begin by defining “system.” According to Merriam-Webster, a system is a group of objects, ideas, principles, or elements that work together for a common purpose. It is an organized network or arrangement that carries out a particular function. When we say that language is a system, then, we mean that it involves a group of elements working together to serve a purpose—namely, communication.
Language, however, is made up not just of individual elements, but actually of several subsystems. Each of these systems contains an organized network of elements in its own right—elements that work together to serve a purpose and accomplish an objective. Linguists identify several subsystems of language, including phonology, morphology, lexicology, syntax, semantics, and pragmatics. We'll look briefly at each of these.
Phonology is all about the sounds of a language and how they are combined and organized into meaningful strings. Linguists that study phonology will be interested in why vowels make different sounds in different words, for instance. Morphology studies how words are built from morphemes (units of meaning). Scholars interested in morphology will identify various grammatical structures and how they combine to make meaning (like -ed or -ing being used to form the verbal past tense and participle, for example).
Lexicology pays attention to forms, meanings, and uses of words as well as their development and history. Syntax studies how words are formed into phrases and sentences and why they are structured and ordered the way they are (why subjects usually precede verbs in English, for instance). Semantics is all about the meanings that are expressed through words and syntax. Pragmatics looks at how language is used in context and how meanings can vary or be revealed or concealed according to a speaker's situation.
These systems, then, combine to make the larger system we call language.