Often used to describe a literal sprint or action to catch something. The policeman had to run after the thief who snatched the purse.
Pro Tip 2/3
Urgency Implied
Indicates a sense of urgency or need to catch up with someone or something. He had to run after the departing train as it was his last chance to get home.
Pro Tip 3/3
Immediate Response
Used when immediate action is required in response to a sudden event. Upon seeing her toddler wander off, she ran after him.