What is the method for growing plants and raising ﬁsh in the same system?
I believe you may be referring to aquaponics or aquaculture. See below.
Aquaculture is a system where aquatic plants or organisms are raised for human consumption. These organisms may be raised in artificial systems entirely created by humans or they may be systems put in place in ponds, coastal waters, rivers, etc. While not all aquaculture systems include plants, some do.
Aquaponics is a closely related term for systems where aquaculture and hydroponics are combined. Hydroponics refers to systems where plants are grown without soil. In aquaponics, both fish and plants are grown.
The diagram below shows a backyard pool converted to an aquaculture system:
*Note, you could definitely argue that the system above is an example of aquaponics too; this is particularly the case for the mint and the taro being grown in water and gravel.
The diagram below shows an aquaponics system: