Manufacturing is the speed at which you can build stuff. Each point of manufacturing cost 1$ to use it, so if you have 10 shields you can spend 10$/turn to build the item. I believe, however, that the 10$/turn is reduced by your social / military spending rate. So if your spending is only 50% you're actually only progressing 5$/turn towards the project (I think, though I haven't quite confirmed this).
Increases to Manufacturing increase your available manufacturing as a %. If you have manufacturing 10 and you build a +10% social project you're now at manufacturing 11, allowing you to build stuff faster.
Some of the manufacturing boost you get is free, it counts as a $1 spent towards the project without costing a $1 from the budget. This is the number after the "+" in the display. So 10 + 2 means your maximium manufacturing is 12, but it only costs 10.
Theres an article in the Encyclopedia that explains this part better.