The matchmaker visiting times are 10:30 AM, 3:00 PM, and 7:00 PM. She will visit at these times for three days, begining a couple of days after your Tamagotchi becomes an adult.
When she appears on screen, press a button. Her suggestion for a mate will appear on screen for a moment. After that, you have the option to accept or decline that partner. If you accept you'll be on your way to the next generation. If you decline, the matchmaker will go away until the next scheduled visit time. If you decline and it's already the last scheduled visit, she won't come back.