They like me! They really like me!
Rapport is a social skill that's utilized when you're trying to get along with someone. This might be to establish a new contact, get on someone's good side, warm them up to doing something you want them to do, flatter a character, or appear to be knowledgeable of a particular set of customs. This skill is opposed by Empathy if there is an active opposition, such as if your target is able to become aware of your intentions. However, it's possible, especially against unnamed NPCs, that a passive opposition will be used based on the character's mood and cunning.
There might be instances where someone tries to tarnish your good name. In most cases, this type of slander is opposed by Rapport, and success can either conjure up contradictory rumors or allow those that know you to disbelieve or discount the tarnishing words. The only time you might use a different skill to battle rumors is when you're trying to maintain a notorious reputation, in which you could use Provoke (or Deceive if appropriate), but unless you have a special skill this method will affect only those present to witness the intimidation, narrowing the results considerably. To preserve your "street cred" on a large scale you can still use Rapport as it's the skill used to manipulate how others see you.
It might seem obvious, but Rapport would be a good skill to use along with other social skills. For example, if you've disguised yourself as someone your target knows there is a chance of creating an advantage to a Rapport skill roll. Likewise, if you're trying to get this target to do something for you, a good Rapport would create an advantage for a Provoke skill roll