manufacturing is very limited, only to making buildings and ships, production affects manufacturing, research and wealth.
manufacturing is divided into three types, social (planet only), ship construction (ships) and all (both)
anything that boosts manufacturing will increase how fast your planet constructs new buildings or upgrades, cranks out ships ect.
production affects research, wealth and manufacturing. typically you get some from population, some techs boost it, and asteroids. the wealthy trait gives 1 production per planet and precursor planets typically also give a bonus to production. there is also at least one mercenary that grants +5 production, and many colonization events can affect it as well. each point of production gives 1 point in all 3 areas, so 4 production means 4 research, 4 wealth and 4 manufacturing.
currently population yields its square root in production, so 4 pop = 2 production, 9= 3, 16 pop = 4 production ect. lately i have taken to placing starbases for asteroid access since having 3 or more asteroids assigned to a nearby planet (max is 1 production, you get less if farther away) it can quickly boost up a planet.