Before it became common for internet access to be available in a private home, only large organizations like universities and governments etc. . . had internet access, which was prohibitively expensive. The bulletin board system flourished because it cost nothing to a user equipped with a modem. A system operator, commonly known as a SysOp, set up a bulletin board system by configuring his or her PC with software to automatically receive incoming calls from other modems, allowing other users to connect with the system. Keep in mind that unlike dial-up internet access, using a BBS was absolutely free provided, the BSS was located in a local calling area and the SysOp (as many hobbyists did) declined to charge a fee for access.
As it name suggests, the primary purpose of a bulletin board system was social - the heart of every BBS was the message board system where users could post text messages on any number of topics. Most hobbyist bulletin board systems would be equipped with only a single modem, and so the message board system flourished given that only a single user could connect to the BBS at any given time.
Because of the speed of early modems - 1200 or 2400 baud, usually, which seems positively ancient by today's standards - much of the activity of an early bulletin board system was the exchange of ASCII or ANSI text data, and most message board systems were set up in this way. Although with the popularity of the internet it is less common now for people to access systems purely to exchange messages, the bulletin board system phenomenon was based on a very small community of users and sole purpose of many boards was to post or receive messages.
In a relatively primitive way, some bulletin board systems offered users the ability to play games and download files, and many 'elite' private bulletin board systems existed for users to exchange copyrighted software. Some larger BBSs had multiple 'nodes' or computers and modems in different places that allows for multiple connection. These were usually pay services, one of the most popular being Q-Link, operated by Commodore, which can be thought of as an extremely primitive version of something like AOL.
The popularity of bulletin board systems in the 1980s also gave rise, amongst some users, to phone 'phreaking', in which users would manipulate the telephone system in order to make long distance calls free of charge. The primary motivation was because a user had to connect to a bulletin board system directly with his or her modem. Today, we think nothing of connecting to a website on the other side of the world, because the data is transferred in packets through many computers on the way, but in the 1980s a bulletin board system had to be 'called' by the modem directly. So a user who wanted to access a BBS outside of his or her local calling area had to either pay long distance charges for the call or find a way around them, hence the popularly of phone 'phreaking.'
Today, the bulletin board system has all but died, being painfully inefficient when compared to the internet. Besides nostalgia, there is simply no reason for an average user to go through the hassle of using a bulletin board system, which is extremely limited by comparison. The influence of bulletin board systems, however, remains profound, and many things that are taken for granted today on the internet were born of the BBS phenomenon.