In linguistics, register refers to the variety of a language used in a particular social setting or for a particular reason. For example, formal grammar is more likely in a job interview than in a casual setting. In a formal setting, people might choose words such as "child" or "father" instead of "kid" or "dad." Other situations that might affect one's language include speaking to a child, being in a legal setting, or being in school. Register refers to the use of language rather than specifically to variables associated with the user (such as age, geographical background, class background, and so on). Registers are affected by the field of an event, including its purpose. Registers are not distinct from each other but instead form a spectrum with no firm, discrete boundaries. In addition, registers are influenced by (and overlap with) other factors, such as dialect and age, which determines one's language choices. 

Register is a part of sociolinguistics, which is the study of the way in which we use language in a social context. MAK Halliday (whose linguistic theory is discussed in the link below) developed and expanded this concept during the 1960s and 70s. To best understand the concept of registers, here is an example:

You probably use multiple registers without realizing it. For example, if you are writing a formal academic paper, the language you use is different from what you might post on Twitter or some other social media. This is because you have audiences with two different expectations about your language and you know how to adjust for these audiences. 

Formal language might sound strange on social media, not to mention being too wordy for the Twitter platform, and informal language will get you a bad grade in academia. 

Language is not just a set of things (words in sentences) that make the same meaning all the time; the people involved in this meaning-making also influence how something gets comprehended. Part of learning language is learning when to use which kind of register.