People’s speaking styles are all the ways that they adjusts their speech to fit different situations.
In formal situations, like at church or in court, many people would speak in a fairly formal style— they would use little or no slang, speak clearly and not mumble, or use more formal forms of address, like ‘Sir’ or ‘Ma’am’. In less formal (more informal) situations–like at a barbecue with friends–their speaking style will be less formal. So, the formality of people’s speech usually reflects the formality of the situation they’re in. But, where it gets interesting for linguists is that more formal speech tends to reflect standard features of a language while less formal speech tends to reflect more non-standard or dialect-specific features of a language. By studying how people change their speaking style, we gain a window into how they think about standard vs. dialect forms.
Beyond such differences between “standard” and “non-standard” speech, other properties can also define a speaking style– anything that somebody does that is distinctive about a certain way of speaking is a “speaking style”. People may have a number of distinctive styles that they use from time to time and using these styles will help them to convey certain additional meanings about themselves, about their attitude toward the situation or the topic of their conversation, about the other speakers, and so on. Speakers and the people they’re speaking to, rely not only on the meaning of the things we say, but also the images and contexts they evoke.