The BBS systems are not popular anymore. There are some still active which you can access via telnet (e.g.:
http://paradisebbs.ath.cx/bbslogin/ ).
Back in the BBS times ANSI/ASCII art was a real big thing, since it was the only way demoscene groups could beautify the bulletin board systems. Again ASCII/ANSI art shows the real demoscene spirit. Having limited resources (just characters in this case) but the wish to produce art.
I ran myself a BBS system back in the 90's. I coded my own system on the amiga. It was quite a fun, but a rather expensive hobby, esp. as a pupil. Since the internet as it is now did not exists, BBS was the only true way for demosceners to spread their productions and cracks worldwide. Most BBS systems were rather closed systems to which you gain access only if you answered certain questions right or a trustful member recommended you.
Technically, it was just a programm which waited for an incoming call. If there was a call the BBS programms sends a command to the modem to accept the call and to establish a connection (handshake). When the connection was established it just sends out text and waited for input (like a shell). That was basically it.
When you wanted to call a BBS you called a direct phone number, ie. if a BBS only has one phone line, only one person at a time was able to connect to a BBS. Bigger BBS systems offered multiple phone lines. The bulletin boards/message boards where sometimes connected between multiple BBS systems and created therefor a net. Normally one BBS system makes a call during the night to another BBS system and exchanged messages.
All I can say that I really did love the BBS era. It was so undergroundish and geeky and therefor cool 8-)