Q: Trying to get free app but asks to verify card details

You can select None for payment method.

Instructions here > iTunes Store: Changing account information

If, "None" is not available, read here >  Why can’t I select None when I edit my Apple ID payment information?

You don't have to have a credit card but it is normally used at some point to validate your address so that, for example, you don't try to access the store of a different country. If you have one but cannot get it verified then there is usually some minor discrepancy between the details that you are entering and those that the institution has on file for you - check against a statement. You may have trouble changing the payment method to None while verification is outstanding.

There are normally some hoops you can jump through as you create your iTunes account that workaround the need for a card, otherwise once the account is set up iTunes may still expect you to provide one, which you can subsequently remove. If you don't have or want to use a credit card at this time you can try this method...

1. Enable the menu bar if needed with Ctrl+B (Applies to Windows only)
2. Sign out of the iTunes Store from the Store item on the menu bar
3. Go to iTunes Store home page
4. Go to Quick Links > Redeem
5. Sign in with your Apple ID
6. Go through the screens, accepting the EULA, etc.  You should arrive at a Review Payment screen where one of the options is None - select this and continue. You shouldn't actually need to redeem a new voucher/card.

tt2

both Carolyn and tt2 are wise... I always refer folks to:

Create an iTunes Store, App Store, or iBooks Store account without a credit card or other payment method - Apple Support

[Existing Apple ID] - What do I do if I do not have a credit card?