What's co-supervisors duty?
This might differ from country to country, but in the country where I did my PhD, only full professors are allowed to confer PhD degrees. In this case it is common that the co-supervisor (who can be on the level of an assistant or associate professor) is repsonsible for day-to-day supervision of the student and often also for coming up with the research project. In this situation, the advisor would usually be less involved in the research itself, but mainly on an administrative level (keeping track of overall progress, authorising travel expenses and the like). He will maintain overall responsibility.

Another reason to have a co-advisor might be to bring in specific knowledge into a project. This is particularly relevant for interdisciplinary research projects.