You can save user details in SharedPreference. Only concern should be security of passwords if you are saving it. Your application's shared preference is sandboxed by default, so your data is safe from other apps.. But a determined/inspired hacker with root access can get any data from any app in an android phone.
For password
1) Either you can use a web service, and store the password in server. During every login process, you can send the username/password to server and validate. This is the best approach if your application is using internet. This option has the simple advantage that you are not saving sensitive data on phone itself.
2) Other option is to store password encrypted. You can use this option if your app doesn't use internet one bit, and you are not ready establish a server for authentication process. There is no absolute security in Android, but saving encrypted does boost the security level.