I regularly get reports from people having issues logging into Reddit using the app.
Unfortunately, the login page is hosted by Reddit, and is outside my control. The page constantly has issues, and is incompatible with many devices.
If you have trouble logging in, please report this to Reddit directly.
Here are a few suggestions which may improve the situation for you:
The system WebView is used to display Reddit's login page. Updating it may fix any issues.
On many devices, this can be updated through Google Play:
https://play.google.com/store/apps/details?id=com.google.android.webview
Users running old Android versions are advised to upgrade to the latest available version.
If you're using mobile data, try WiFi, and vice versa.
Some users have reported that having a password manager installed (which auto-fills your Reddit password) can break Reddit's login page. Try disabling this and entering the password manually.
If you are proxying traffic through an adblocker (or any other application that could block or modify web resources) please disable this during the login process.
Reddit often experiences outages. When this happens, the site may be viewable by logged-out users (as it's cached in Reddit's CDN), but actually interacting with the site is broken for everyone. See the following link to check for known outages:
Some users have reported that Reddit's OAuth login site fails to accept passwords that have special Unicode characters in. I've reported this to Reddit but they can't reproduce the issue. In any case, it may be worth trying changing your password to something that only uses simple characters (A-Z, a-z, 0-9).