Generally, how it works is this:
'Vacation Days'/Paid Time Off (PTO) - these are days that the company will pay you your salary, as if you are working - but you aren't working. In the US, I don't believe there is any minimum mandated, whereas in other parts of the world there is minimum number of days an employee is entitled to a year.
Also these aren't awarded in bulk, they are generally accrued at a set rate e.g. you 'earn' 1 day off for X number of days worked.
Now, at any time (we'll get to this in a minute) - you can take leave off and have it as unpaid leave. This can be either you don't have any PTO, you've used it all up or you haven't accrued any PTO yet.
In your case, since it's a summer job and you've indicated you are fine taking unpaid leave - here is how it works:
You put in a leave request for the number of days that you are going to be absent with a start and end date and that it's unpaid leave, your employer then approves the leave.
There are some caveats - in most cases and employer with have a policy about how much notice to give - this varies, but typically the longer the leave requested, the more notice you are either required to give or ought to give. 10 days isn't unreasonable, but it is significant - so I'd be giving them at least 1 months notice. Since you've indicated it will be over the summer - if you have dates, best to give them the dates now. Pro-Corporate Life Tip - Send them the request with the dates via email and never accept a verbal confirmation, ask for confirmation in writing This will help if they try and pull a fast one later on or claim it was never approved.
Generally, unless your boss/workplace is toxic, anything over 1-2 months worth of notice should be automatically guaranteed - the only exceptions would be if the time that is being requested off is a peak time or there is a project that is needed etc. etc.
They might grumble a little - but having a 2 month notice that you are going on unpaid leave for 10 days shouldn't be difficult and since it's a summer job, I'm guessing that they often use teenagers/people on their first job - so they are probably used to it (it might be why the HR person is a bit stern, used to kids pulling stuff whilst they learn what it means to work).