The Component Assignment gives you the ability to assign your box to parts of your Joomla website where a third-party extension is handling it. For instance you could select the popular K2 Extension and have the box be displayed at all the pages created by K2.
In the table below you can see which options are available to setup the Components Assignment.
Click in the textfield so a selectbox pops up from which you can filter the 3rd-party extensions that you'd like by typing on the fly.
Once you select one you can continue typing to find the next one and so forth.
A very common use case is using the Component Assignment for occassions where the third-party extension is very particular to its purpose and so does the content of your box. For instance, if you want to show something related to newsletters then it can be a wise choice to choose the popular AcyMailing Joomla Extension.
It is important to note that the list of components that you can choose are limited to those who actually have a frontend. Strictly backend components are naturally not available for this assignment as the box can not be displayed on the backend.