When chlorine (as a gas or dissolved in water) is added to sodium bromide solution, the chlorine takes the place of the bromine. Because chlorine is more reactive than bromine, it displaces bromine from sodium bromide.
The solution turns brown. This brown colour is the displaced bromine. The chlorine has gone to form sodium chloride.
In this equation, the Cl and Br have swapped places:
This type of reaction happens with all the halogens.