I made an IRA contribution in Jan. 2018, but I forgot to claim it on my 2018 tax returns. Can I claim the contribution on my 2019 tax returns?

[macuser_22]

What type of IRA - Traditional or Roth?

If Roth then there is nothing that requires amending.

A Roth IRA contribution does not actually go on a tax return, but you should enter it anyway to:

1) Tell you if your income qualifies you for a contribution and warn you if it does not.
2) Check if your income exceeds the limit to contribute to a Roth.
3) Track your contribution year-to-year if you use TurboTax every year.
4) Add the Retirement Savers Credit if you qualify.

If Traditional then it must be reported on a amended 2018 tax return to report it and claim a deduction if yiu want to deduct it.

If it is a non deductible contribution then you need to file a 2018 8606 form reporting the non-deductible contributor.