The Simpson family debuted in short animated cartoons on The Tracey Ullman Show for three seasons before being spun off into their own half-hour series. These shorts, also called "bumpers", aired before and after commercial breaks during the first and second seasons of the show. They eventually had their own full segments in between the live action segments during season three. They did not appear in the fourth and final season, as they had their own half-hour TV series by then.
All of them were written by Matt Groening and animated at Klasky-Csupo by a team of animators consisting of David Silverman, Wes Archer, and Bill Kopp. Tracey Ullman Show cast members Dan Castellaneta and Julie Kavner provide the voices of Homer Simpson and Marge Simpson respectively. In the beginning, the drawings appeared very crude because the animators were more or less just tracing over Groening's storyboards, but as the series developed, so did the designs and layouts of the characters and the "Simpsons drawing style" was ultimately conceived. This style evolved even more throughout the first few seasons of The Simpsons and was used more than a decade later on Futurama, another animated series created by Matt Groening.

There are 48 Simpsons shorts on Tracey Ullman Show, that you can watch below. I recommend you to do so, because some of them are very good and funny, even if the animation is not a very good one - old is gold.