I qualified for hope credit and other educational credit, do I need to fill out an application?

The Hope Credit was replaced by the American Opportunity credit, several years ago.

There's a new urban myth among college students that says they can get a $1000 from the government just for filing a tax form. For most of them, they simply aren't eligible. A full time unmarried student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on parental support, 529 plans or student loans & grants. You usually must have actually paid tuition, not had it paid by scholarships & grants.  It is usually best if the parent claims that credit. 

You cannot claim a credit if you are, or can be, claimed as a dependent by someone else.

If you are eligible, you just follow the TurboTax (TT) interview*. TT will fill out form 8863 (there is no separate 'application", form 8863 is an attachment to your tax return). But, a lot of people are just not eligible. See https://www.irs.gov/Individuals/AOTC for full rules 

 or https://www.irs.gov/individuals/llc

*In TurboTax (TT), enter at:

Federal Taxes Tab (Personal for H&B version)

Deductions & Credits

-Scroll down to:

--Education

  --Education Expenses