Bronchitis is a bacterial infection of mucus within the sinuses and lungs. The virus that caused the initial congestion as well as the bacteria causing the bronchitis can be spread by someone coughing and sneezing.
Yes, bronchitis is very contagious. Bronchitis is catching through coughing and sneezing. You shouldn't be exposed to someone with it until all fever is gone.
Bronchitis that is caused by infection by either bacteria or, more commonly, by viruses are in fact contagious. However, there are inflammatory non-infectious causes which are not infectious. These are generally less common.