Creating a Section Aware webpart in SPFx

Recently, I built a web part for a client, which led to a discussion about why the web part background was static white, which did not reflect the branding on the page. My quick fix was to just change the color manually, but now I wanted to know more about how I could build webparts that are aware of the area that they are in. It turns out, there are several options, depending on the capabilities needed and the web part framework.