Comment depth refers to the nesting of comments, whereby comments are followed by replies, which are typically indented so that they make a threaded discussion easier to follow.
However, not all threaded, or nested, discussions are truly easier to follow, since a lengthy or complicated discussion can lose sight of the parent comment that triggered it in the first place. There gets to be a point where nesting goes from nice to nasty. Where exactly that point is, is totally up to you and your audience.
Within the WordPress admin panel, you can change the nesting depth by going to:
Settings -> Discussion -> Other Comment Settings
Then changing the value in the dropdown:
Enable threaded (nested) comments _____ levels deep
No matter what maximum depth setting you choose, should someone reach the max level and post a reply, it will not be associated with the thread it was intended to be part of - creating an all new thread of comment on its own. This is by design and is just something of which to be aware.
If you’ve got a lot of comment/reply activity and your settings are to paginate comments, keep in mind that you can lose track of much of the conversation when hopping page to page, just to see how the conversation unfolds. For this reason alone, many bloggers switch to lazy loading, without pagination. It can improve the user experience as well as the page load times, which is good for users and search engines alike.