I filed my taxes before my dad; he told me he could claim me as a dependent, but when he tried to file, it turned out I earned too much. How will that affect my return?

Were you a student last year?  How much did you make and how old were you on December 31, 2018?

If you were a full time student under 24 it doesn't matter how much you make, he can claim you.
You can amend your return to claim yourself and see if you will get any more refund.  Or did you get back all your withholding?  How much did you make?  The amended return worn't be ready until Feb 28.

I was a full-time student. I graduated in May and turned 24 in October. I don't know if I got back all of witholding (first time filing alone). I made a little over $21,000 over three jobs.

Ok you were 24 in 2018 and made too much so he can not claim you.  Yes you should amend.  You might get more refund.  

How to amend an Online return.  The Amended return is delayed  https://ttlc.intuit.com/replies/7159321 https://ttlc.intuit.com/replies/3288565

To see if you got back all your withholding......add up box 2 on the W2s.  Does that match your refund?  

You will owe tax on income over 12,000.  The Standard Deduction for Single is 12,000.  Which means the first 12,000 of income is tax free.