What is the difference between pushing down search results (suppression), de-indexing, and removal?
Search results can be removed, or they can seem to be removed. When a search result is removed it is often because the article itself has actually been removed from the source. But there are ways to remove search results from Google and still leave the result on the original web page, this is called NoIndexing.
Delete the Page
True "removal" removes the whole page. It's deleted by the webmaster. This is the most effective way to make search results disappear because search engines will come by, discover the page is gone, and automatically delete the search result. This has to be done by the publisher (usually the webmaster).
NoIndex Search Results
The NoIndex tag is the online equivalent of painting lambs blood on your door. NoIndexing is when a webmaster adds a tag to a page without taking it down. Instead of removing the page, they simply add a tag to it. That way the page can still be read by people using the site, but doesn't show up in search. The tag goes in the HEAD area of the HTML code and looks like this:
<meta name="robots" content="noindex">
This small but powerful tag makes a search engine pass over a web page and ignore it.
Remove Search Phrases
If a webmaster is willing to remove search phrases the page may disappear from search results, or simply fall back to where it is seldom if ever seen. This is because it becomes "less relevant". It is still possible for it to rank for a specific search phrase though. Googles artificial intelligence system RankBrain, a part of the Hummingbird algorithm, may still find a page relevant even if it doesn't have the exact search term on the page.
Removal from Googles Index
De-indexing Google or Bing results is the internet equivalent of removing your house from all maps. It is different than NoIndexing. With de-indexing the site is removed from the search engine index, but not from the source web page. Both NoIndexing and DeIndexing have the same effect though - both remove search results from Google or Bing (or Yahoo).
Suppression pushes search results down, "replacing" them with better more relevant ones. During a suppression campaign online reputation companies manage the development of very specific web properties common to similar entities (people, brands, etc.). For example, Doctors and Dentists tend to return search results very different from Attorneys. Reputation firms design web development and content plans to fill gaps in search. Then apply search marketing techniques to promote the right search results.