What are negative keys used for? Somewhat new to using standard SQL databases (currently working with MySQL mostly) I haven't run across many usages of this as of yet. 

When and why is it useful to have negative (or rather signed) keys indexing a table?
All a primary key is is a value that we have determined is the value that is of utmost importance in a record. Whether that key is a signed int, an unsigned int, a string, a blob (actually, there are limits) or a UUID (or whatever name it takes today), the fact still stands that it is a key, and that it is the thing of utmost importance.

Since we're not constrained to use only positive oriented numbers for our keys, it makes sense to consider that a signed int will only go to ~2 billion, whereas an unsigned int will go to ~4 billion. But there's nothing wrong with using a signed int, setting the initial value to ~ -2 billion and setting an increment of one. After ~2 billion records you'll hit "zero" and then you'll continue to ~2 billion.

As to why it would be helpful to have "negative keys" in a table, that's the same question as "why is it helpful to have keys in a table". The "value" of a key has no impact on its status as a key. A key is a key is a key.

What is important is if the key is valid.

As to why it would be useful to allow keys that were negative, I can suggest some reasons:

What if you wanted to indicate returns in a sales system as negative sales order numbers, that matched the positive sales order number, thus making correlation easy (this is naive, and poorly designed, but it would work in a "spreadsheet" sense).

What if you wanted to have a users table, and indicate that the ones with negative numbers were system controlled (SO does this very thing, for chat feed users).

I could go on, but really the only reason why the number being negative is of importance is if you or I assign importance to it. Aside from that, there is no great reason for the value of a key to have any bearing on the key itself.