I thought of a list of things that would have never happened if Katherine wasn't here, and I came up with a good bunch. 1. Without Katherine, Stefan and Damon would have stayed human and died a long time ago.
Katherine made Stefan die and become awesome, and with that he turned the Damon we all amor into himself. So she practically gave us the hot, badass Damon we all adore so much. 2. If it wasn't for Katherine, Elena wouldn't have been even born.
If Katherine didn't have a Baby and that baby didn't have a baby and that baby didn't have a baby etc. there would have been no Elena would there? 3. If Katherine wasn't there, Caroline would have stayed her old human annoying self and never been that awesome.
She would have never even turned Caroline and most of us would still hate her like we did in the beginning. 4. If Katherine didn't compel Matt Tyler would have never become a werewolf.
If she did not compel Matt to fight Tyler until he kills him, he would have never went to that room and got Tyler drunk so he could fight him. And he never would have killed that girl. 5. Last but lot least, if there was no Kat, Elijah, Klaus and Rebekah would have been looking for a totally different person and we might have never even known they existed.
They would have been looking for another Petrova that might have no relationship with Damon, Stefan, Elena whatsoever.
So Katherine pretty much makes up the whole show, and THIS is one of the biggest reasons I personally amor her.