Can someone explain the prons and cons of DSLRs vs Mirrorless cameras?

I haven't owned any camera other than a point and shoot. I'd like to step up and get a better camera. I am a casual photographer and will mostly be shooting pets and such indoors and occasionally outside when we go on vacations and stuff and also would like the camera to take good pictures in bad light and at nighttime outdoors.

I am looking at entry level DSLRs and Mirrorless cameras. Can somebody explain in layman terms the main difference between the two types of cameras and which might suit me better for my use case? I want a camera that takes fast pictures and also prefer one that has both a view finder and also an LCD display. Do both the DSLRs and Mirrorless cameras have that? I like the small form factor of the mirrorless cameras, but I am not sure what I am giving up in features or picture quality by going with the small body. The Canon Rebel S1 might also be an option, given its small size.