During this time period, there were a number of different unions with different demands. Some, like the Knights of Labor, wanted to completely reform the system and give the workers ownership of the factories. Others, like the AFL simply wanted better treatment within the system.
Overall, though, the basic demands were for better working conditions and higher pay. For example, the National Labor Union formed in 1866 to push for a law mandating that workers would not have to work more than 8 hours per day. The Knights of Labor and other unions wanted higher wages. The AFL wanted laws mandating workplace safety. All of these are examples of the sorts of demands for better pay and better conditions that were made by unions in general during the time you mention.