Mother 2
In Mother 2 Tessie appears in Winters. She helps Jeff and the Bubble Monkey get over the lake to get to Ness and Paula. She has a club called the "Tessie Watching Club" who have different opinions on where she lives. One of them says that Tessie is also titled the Lake Tess Monster, which is in reference to the Loch Ness Monster. She is found in the water, and she looks like a big blue dinosaur. She will take you on a ride to the area right before Threed. This is the only place that Tessie will escort you.
 Mother 3
In Mother 3 Tessie is a normal character living in Tazmily Village. She has no major role in the game, or any role at all. She is just another towns-person.
Tessie from Mother 2 will appear in a museum at the end of Mother 3, along with a lot of other stuff from Mother 2.