Stop + gerund means that an action in progress ends. There is no causal relationship between the two verbs.
- I stopped working for them last year.
- I have stopped cycling to work.
- Could you stop bringing your dog to the office?
- He stopped eating meat in January.
Stop + infinitive means to interrupt an activity in order to do something else, so the infinitive is used to express the purpose of stopping.
- I stopped to have lunch at 12:00.
- It's difficult to concentrate on what you are doing if you have to stop to answer the phone every five minutes.
- We stopped to look at the wildflowers.
- I stopped to take a break, but I'll start again soon.