There were a bunch of predefined modes in the video BIOS, and with a little bit of assembler you'd issue an interrupt (a system call really) which would change the video mode. Then as the parent comment said, you could write to video memory directly and your writes would either be interpreted as ASCII character/attribute pairs in a text mode, or colour palette indices in a graphical mode.

Most games at that time used mode 13h which was 320x200 with 8-bits per pixel which therefore indexed into a 256-colour palette (which could itself be redefined on the fly via reading from and writing to a couple of special registers - allowing for easy colour-cycling effects that were popular at that time). Here's a list of the modes: https://www.minuszerodegrees.net/video/bios_video_modes.htm