To make a scene means to create a dramatic situation in public, to show too much of emotional display or unnecessary attention, which often makes people surrounding uncomfortable. For example to have a loud argument at a restaurant.
You can also say to cause a scene or create a scene.
✦ “Barrack was hoping to ride all the way to school with the girls, but he knew it would create too much of a scene.” – Becoming, Michelle Obama
(Barack was afraid that his heavy vehicles, his staff and security coming along would draw too much attention and a noisy situation for his daughter’s school.)
✦ Please don’t make a scene here. Let’s settle our problem privately.
(Please avoid talking about our problem in front of many people.)
✦ I’m sorry for making a scene, I was so mad and couldn’t control myself.(Sorry for showing too much of private emotions in public.)
✦ I’m breaking up with you and please don’t make a scene.
(I know it will hurt you but please don’t embarrass us here with your negative feelings.)