If you are already logged in on a mobile/desktop browser somewhere
You can check your profile, the "Edit" tab across the top. From there click the "My Logins" option along the side. That should tell you what login method your account uses.
The Stack Exchange network does not have its own login system. Stack Exchange does have its own OpenId provider, however, that you may or may not have set up when you signed up. If you didn't, then you likely validated through another login service, such as Google, Facebook, Yahoo, or other similar providers.
Once you know which method you used, you will be able to login using that method from the login page.
No, you don't need to change your username for this, though you may if you wish. That option is also under your "Edit" tab on your profile, and may be different per site you have an account on if you wish. Your username is not used for any login method, even the Stack Exchange OpenId method, and is only for display. It does not even need to be unique.
If your only logged in instance of your account is on the app
This is more difficult, as you can't edit your profile from the app, but you do still have a couple options. I don't have the iOS app myself, so any app specific instructions will be for the Android app. If the instructions are different for the iOS app, someone with that app can either comment or edit in the iOS instructions.
1) Perhaps the easiest way to find out what you logged in with is if you've gotten emails from any of the sites you have accounts on. If you have, that can help narrow down what login method you used. While it could still be another method that gives your email to Stack Exchange, such as Facebook, it could be that you used that email provider directly as your login method. Try those methods.
2) Log out of the app, by going to the menu then settings and "Log Out", look at the various login options, and pick the one you feel you would most likely use. This method has a chance of locking you out of your account, however, so use it only if you truly feel you could figure it out that way.
3) Try to login on on of the sites anyway. Pick a method you're most likely to use and try to login with that. If it doesn't work, you may end up creating a new account with this method. If you do end up with a new account from this method, you may then use the "merge account" process to merge the two accounts, and therefore give you access on a desktop/mobile browser to your profile.
Note: If you end up with two accounts and try to merge, you should get an email sent to both accounts. If they have the same email, even if they use separate logins, then that shouldn't be an issue. However, if they have separate emails (due to you having separate emails, for example) then you'll need to check each email address you have to verify where the other email was sent. If it turns out to be an email you no longer have access to, you'll have to explain to the team and they will go from there.
If you don't have any emails you have lost access to, this will likely not be an issue for you. Just be sure to check your spam folders as well to make sure you got the emails.