Is staff plural?
- The family is on a tour
- The family are on a tour
Which one is correct and why?
Both are correct, as the collective noun family can both be treated as singular or plural. However, one is less likely to hear the second sentence in the US, for collective nouns are usually treated as singular in American English, while they can be treated as both singular and plural in British English. See the note on collective noun concord at Oxford Dictionaries Online.
You may want also want to consider a related answer here.