I am 21 year college student whose parents have always claimed me as dependent. I had a summer 2019 internship making $14,000 and also paid some tuition. How do I file for taxes/get a refund?

Yes you have to file your own return.  If you are under 24 and a full time student your parents can still claim you no matter how much you make.  On your return you need to check the box that says you can be claimed by someone else.

You will get a W2 at the end of the year in January from your employer.   Just enter it into Turbo Tax exactly as shown.  

So I am a student in New York City that is 21. My parents have always filed joint with me as dependent (I think, I don't understand any of this jargon/terminology).

Only married people can file a joint return.  Your parents claimed you as a dependent on your parent's joint return. (Assuming your parents are married to each other and did file as married filing joint, as opposed to married filing separate.)

This summer I had a finance internship in NYC where I made approximately $14,000 and walked home with about $9800 post tax (I think, I don't remember). I'm really confused on how exactly taxes work and what things I need to or what forms I need.

Whoever the employer/contractor was that hired you will send you the appropriate tax reporting documents at tax filing time. They have until Jan 31, 2020 to put them in the mail to you.

Is 14,000 enough income to file for taxes? Or did I make too much money and have to file for my own taxes? How do I file for taxes? Do I file with my parents or by myself?

There is no monetary limit that would would refer to as "enough". You could earn $1 and file a tax return if you wanted to - though for $1 you wouldn't be legally required to.  Basically, as a single person if you earn less than $12,000 during the tax year then you are not required to file a tax return. However, if the employer withheld taxes on that income then you might "choose" to file a tax return in order to get those withheld taxes refunded to you.

Also, I recently just paid $868 dollars in tuition out of pocket.

The deduction for out of pocket qualified education expenses expired at the end of 2017. Congress did not renew the deduction for the 2018 tax year, and chances are they won't for the 2019 tax year either. But if they do renew that deduction, you don't qualify to take it. That's because you *QUALIFY* to be claimed as a dependent on your parent's tax return. It doesn't matter if they actually claim you as a dependent or not. If you "QUALIFY" as their dependent, then you don't qualify for the deduction.  (Unless congress were to change that in the law, of course.)

 How can I reflect my out of pocket tuition expenses?

Unfortunately, the way the laws are on this right now, you can't.

What documents or items do I need from my employer

Depends on if you were hired as a W-2 employee, or a 1099-MISC contractor. But either way, they are required by federal law to send your your tax documents by Jan 31, 2020.

Most importantly, how do I maximize my refund?

You have nothing to "maximize" here. Your first $12,000 of earned income is tax free. The remaining $2K (give or take) will only be taxed at 10% - the lowest tax bracket. The only possible way to reduce it more is to donate that remaining $2K to a qualified charity, and 50% of your donation will be tax deductible.

Now if this is "truely" earned income, you can open a traditional IRA and with $14K of earnings if you can contribute a maximum of $6,000 to a traditional IRA and not pay taxes on that contribution. But first you should read https://www.irs.gov/retirement-plans/retirement-plans-faqs-regarding-iras-contributions to make sure limits below the maximum contribution amount allowed don't apply to you.

As a reminder, when you complete your 2019 tax return in 2020, you ***MUST*** select the option for "I can be claimed on someone else's return". You don't have a choice here. But your parents do have a choice to claim you or not. For you, if you just *QUALIFY* to be claimed on someone else's tax return. (and I'm 100% sure you do) then you must select the aforementioned option on your tax return. (This is assuming you did not graduate before May 1st, 2019. If you graduated on or after that date, that may or may not change things.)