1. To exert pressure on and move in contact with someone or something. The girl I started dating has a dog that rubs up against me whenever I go over to her house. There's a piece of metal rubbing against the tire that's causing it to wear down very quickly.

2. To exert pressure on and move something while in contact with someone or something else. In this usage, a noun or pronoun is used between "rub" and "(up) against." Someone was rubbing their hand up against my back on the train. It made me really uncomfortable. The elephant rubbed its hindquarters against the large boulder to relieve its itching.

3. To contradict or come into conflict with someone or something. The new report, cowritten by nearly two dozen of the world's top scientists, rubs up against the government's official stance on climate change. Some people simply refuse to accept reality when it rubs up against their own personal world views.