The fields for Fandoms, Characters, Relationships, and Additional Tags suggest tags as you enter search terms. Choosing the "canonical" or common tag, (that is, the ones included in the autocomplete list) will return all results containing this tag, its synonyms, and the subtags linked with the tag. For example, picking the canonical relationship tag Jean-Luc Picard/Beverly Crusher will return works tagged Beverly Crusher/Jean-Luc Picard as well, assuming these tags have been linked by a wrangler in the background. Refer to What is a 'canonical' tag? for more information.
If a tag doesn't appear in the autocomplete menu, it doesn't necessarily mean that the tag doesn't exist on the Archive; it just hasn't been marked as common by a tag wrangler. You can enter any word or phrase here. If your phrase doesn't match a common tag exactly, it will find all tags containing the words in your phrase. Entering People Doing Things, for example, will also find works tagged Nice People Doing Things, People Doing Shady Things, and People Doing Things with Spoons. Depending on your search, results in this case will be somewhat unpredictable.
The more search terms you enter or options you pick, the more your search results will be narrowed down. By default, all the search results are AND search results. This means that entering two fandoms will only find works tagged with both, not all works that are in one fandom or the other. Entering two characters will only find works that have both. Picking M/M and F/M will only include a work in the search results if it has both category tags, and so on.
You can find more about tagging in our Tags FAQ, and more about tag search in the Search and Browse FAQ.