Character generators. I learned on a Chyron in college. Basically, the old way would be two different video streams: your main picture, such as an anchor at a desk, and the character (or key) which would be overlaid on top of the main picture. It had its own set of controls that let you fade it in and out, create and store preset keys for your broadcast that you keep in order and bring up as needed, somewhat like a PowerPoint, and you can also create new keys on the fly if necessary.

Many of the fancier-looking graphical effects were made with Scanimate, a commercialized analog computer with functionality to warp and composite images. There were only a few of these systems ever made and a similarly small number of operators(making them astoundingly unique in many ways), but the effects produced with them were nearly ubiquitous in late 1970's and early 1980's TV. In the mid-1980's this kind of custom graphics production rapidly shifted towards digital computers, but dedicated character generator hardware remained in use for considerably longer.

I should note that while Scanimate is a good singular example of the kind of technology being used, graphics production can draw on any number of methods, so studios would also come up with their own techniques as a matter of course.

A classic example of Scanimate warping in Sesame Street

A Scanimate demo reel showing typical effects

The first example used chroma-key (blue-screening, though later green replaced blue in most uses). A dedicated system knew how to replace a particular shade of one color with a secondary image (the map). It was in use by the '80s (source: uncle was a weatherman; visited the set and saw it).