A controller is just as it sounds. A circuit(s) that control a process. USB Controllers are responsible for controlling USB ports, for example.
Interfaces are places where components area connected. A SATA connection could be called an interface. NIC stands for Network Interface Card.
I don't know what you mean my "adapt cards" maybe you mean daughter boards or Adaptec cards? Adaptec is a manufacturer of computer components and a daughter board or riser board are used in servers and some odd form factor desktops to make interfaces available in non-natural locations given the layout of the motherboard.
A bus connects controllers to each other and to the CPU and to interfaces. Data travels on the busses from one place to another. This is how data gets from the hard drives to the RAM to the CPU to the video card, etc.
A chipset is a combination of controllers on the same piece of silicon. There are, traditionally, northbridge chipsets which handle high-speed connections like RAM, CPU and video and a southbridge chipset which handles slower operations like disk I/O, USB, PCI, etc. This is changing in modern designs however and some designs have a direct bus between the CPU and RAM and other major controllers are all being made on a single piece of silicon.
So for an example of how it all works together: When you plug a device like a USB drive into a USB interface, the USB controller that is probably located on the southbridge chipset interacts with the device over the bus. Any data that is needed is sent to the CPU for processing and if anything needs to be displayed on the screen as a result, the CPU will send data to the northbridge chipset which will be delivered to the video card for display to the monitor, which is probably connected to the computer by a VGA, DVI, HDMI, or Displayport interface.