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Yes it differs by company, but normally the highest ranking person would have the final say. But in your example, the HR director would talk to the other person and put pressure on them to agree. Or yes, the HR person could go to the CEO. And at high levels, lots of things happen unofficially.

The actual issues here are important and dont warrant an ETC...

Breaking Rules...would HR be involved based on what rules were broken?  If HR has a legitimate concern for the company and the CFO just wants to keep their top salesperson (or whatever)...that would be a good reason to circumvent the CEO. 

All of that said, it gets tricky.  Just look at Harvey Weinstein and his company...I'm sure plenty of people wanted him gone and I bet all of their HR department hated their jobs...but he was the driving force behind the company...

I will first be informative and teach, and then address the question detail.

There can be many different possible structures in a business. Some are formal, and some are informal. Some are well run and organized and others are dysfunctional. I worked in large global companies that over the years restructured a few times.

A well run and structured company has a formal chain of command.

An employee should only have one direct manager. There could be two if one is administrative and one is operation or functional.

HR is generally a central group responsible for benefits, resource management, people legal issues, and administratively hiring and firing. They can also hear complaints. They normally report to a very high level executive.

A person's manager would report to their manager, an so-on up the line to a Chief Operating Officer (COO) to the Chief Executive Officer (CEO).

Any two employees, at any level, and their chain of command eventually intersect into one person. That one person has the final say unless there is a legal issue brought up by HR, in which case there are two people again, and their intersecting boss, eventually the CEO. It's like branches of a tree or a roots structure leading to the trunk.

In a well run business, it rarely gets to that intersecting bigger boss. The under-levels try to cooperate because they know structures change and the high level guy you disagree with may be your boss tomorrow. The top manager of purchasing knows the

top manager of sales and top manager of production, etc and even though they are in different groups, they know when to work together versus when it's so critical it has to move up higher.

Now, your detail,

Low level employee screwing up.

Certain outside of organization director wants that employee fired.

Employee's direct manager does not want to lose the employee.

The outside director can approach the employees manager and have a private meeting. Various things can result.

1) A very serious warning including "stay clear of that director"

2) Employee is fired. His boss knows how the game is played and won't put his job on the block.

3) Employees boss takes the issue up his chain of command to get instructions. It gets resolved one way or the other by the big shots. The low level employee could be transferred to another job or location.

Yes, HR can intervene one way or the other, and CEO has all final decisions.

Suppose the low level employee has a "sponsor" in the company. I had two, senior VP's in another organizations that had a lot of influence.

This changes situations. If you read everything I wrote, this unofficial connection can be very helpful to have. If I had a problem with my direct manager (which was very rare), I could make a telephone call, and an outside executive sponsor would "speak" with my manager, generally stated as a recommendation.

HR generally only handles paperwork. He/she can privately say "We need that Latino, can you give him/her a break?" Or, they can take the other side, "There are several complaints on file. It's time to let that employee go."

Many issues in well run companies are handled informally, even when the structure is formal for when it is needed.

Everyone knows the unwritten rules.

Don't burn your bridges. You ever know when you'll need an important friend. You never know you you'll be working for in the future. Don't shake the trees unless you have to. Don't get things get to the top. Resolve them between you.

Generally speaking, the highest person on the totem pole who cares enough to get involved will make the decision as to which person below gets to make the decision.  

Unless it's a severe infraction, the employee will be given warnings.  If the behavior doesn't improve than even the manager would be on board at that point.