I use Keepass for this. The android port (https://play.google.com/store/apps/details?id=com.android.ke...) works perfectly fine for me, and it looks like there are a few ports for iOS as well. Keepass itself has, of course, long established itself as a solid password manager. And the cost is $0.
Android has two versions of Keepass. Make sure your read the details before you choose which one you want to install. The difference between the two is how/where you store your key file, either on their server or on the local sd card.
