There are several ways to do this, shown below.

As a friendly warning, be careful and up-front to the user when dealing with account, profile, and contact data. If you misuse a user's email address or other personal information, bad things can happen.

Method A: Use AccountManager (API level 5+)

You can use AccountManager.getAccounts or AccountManager.getAccountsByType to get a list of all account names on the device. Fortunately, for certain account types (including com.google), the account names are email addresses. Example snippet below.

Pattern emailPattern = Patterns.EMAIL_ADDRESS; // API level 8+
Account[] accounts = AccountManager.get(context).getAccounts();
for (Account account : accounts) {
    if (emailPattern.matcher(account.name).matches()) {
        String possibleEmail = account.name;
        ...
    }
}

Note that this requires the GET_ACCOUNTS permission:

<uses-permission android:name="android.permission.GET_ACCOUNTS" />

More on using AccountManager can be found at the Contact Manager sample code in the SDK.

Method B: Use ContactsContract.Profile (API level 14+)

As of Android 4.0 (Ice Cream Sandwich), you can get the user's email addresses by accessing their profile. Accessing the user profile is a bit heavyweight as it requires two permissions (more on that below), but email addresses are fairly sensitive pieces of data, so this is the price of admission.

Below is a full example that uses a CursorLoader to retrieve profile data rows containing email addresses.

public class ExampleActivity extends Activity implements LoaderManager.LoaderCallbacks<Cursor> {
    public void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        getLoaderManager().initLoader(0, null, this);
    }

    @Override
    public Loader<Cursor> onCreateLoader(int id, Bundle arguments) {
        return new CursorLoader(this,
                // Retrieve data rows for the device user's 'profile' contact.
                Uri.withAppendedPath(
                        ContactsContract.Profile.CONTENT_URI,
                        ContactsContract.Contacts.Data.CONTENT_DIRECTORY),
                ProfileQuery.PROJECTION,

                // Select only email addresses.
                ContactsContract.Contacts.Data.MIMETYPE + " = ?",
                new String[]{ContactsContract.CommonDataKinds.Email.CONTENT_ITEM_TYPE},

                // Show primary email addresses first. Note that there won't be
                // a primary email address if the user hasn't specified one.
                ContactsContract.Contacts.Data.IS_PRIMARY + " DESC");
    }

    @Override
    public void onLoadFinished(Loader<Cursor> cursorLoader, Cursor cursor) {
        List<String> emails = new ArrayList<String>();
        cursor.moveToFirst();
        while (!cursor.isAfterLast()) {
            emails.add(cursor.getString(ProfileQuery.ADDRESS));
            // Potentially filter on ProfileQuery.IS_PRIMARY
            cursor.moveToNext();
        }

        ...
    }

    @Override
    public void onLoaderReset(Loader<Cursor> cursorLoader) {
    }

    private interface ProfileQuery {
        String[] PROJECTION = {
                ContactsContract.CommonDataKinds.Email.ADDRESS,
                ContactsContract.CommonDataKinds.Email.IS_PRIMARY,
        };

        int ADDRESS = 0;
        int IS_PRIMARY = 1;
    }
}

This requires both the READ_PROFILE and READ_CONTACTS permissions:

<uses-permission android:name="android.permission.READ_PROFILE" />
<uses-permission android:name="android.permission.READ_CONTACTS" />

There is an Android api that allows the user to select their email address without the need for a permission. Take a look at: https://developers.google.com/identity/smartlock-passwords/android/retrieve-hints

HintRequest hintRequest = new HintRequest.Builder()
        .setHintPickerConfig(new CredentialPickerConfig.Builder()
                .setShowCancelButton(true)
                .build())
        .setEmailAddressIdentifierSupported(true)
        .setAccountTypes(IdentityProviders.GOOGLE)
        .build();

PendingIntent intent = mCredentialsClient.getHintPickerIntent(hintRequest);
try {
    startIntentSenderForResult(intent.getIntentSender(), RC_HINT, null, 0, 0, 0);
} catch (IntentSender.SendIntentException e) {
    Log.e(TAG, "Could not start hint picker Intent", e);
}

This will show a picker where the user can select an emailaddress. The result will be delivered in onActivityResult()

This is quite the tricky thing to do in Android and I haven't done it yet. But maybe these links may help you:

• Android Issue 1073:Google Auth Tokens should be accessible to 3rd party applications through an API
• SDK API AccountManager in Andriod 2.x+

Add this single line in manifest (for permission)

<uses-permission android:name="android.permission.GET_ACCOUNTS" />

Then paste this code in your activity

private ArrayList<String> getPrimaryMailId() {
    ArrayList<String> accountsList = new ArrayList<String>();
    try {
        Account[] accounts = AccountManager.get(this).getAccountsByType("com.google");
        for (Account account : accounts) {
            accountsList.add(account.name);
            Log.e("GetPrimaryMailId ", account.name);
        }
    } catch (Exception e) {
        Log.e("GetPrimaryMailId", " Exception : " + e);
    }
    return accountsList;
}

The suggested answers won't work anymore as there is a new restriction imposed from android 8 onwards.

more info here: https://developer.android.com/about/versions/oreo/android-8.0-changes.html#aaad

For Android 8 and above -

Step 1 - Add the following code in AndroidManifest.xml -

<uses-permission android:name="android.permission.GET_ACCOUNTS" />
<uses-permission android:name="android.permission.READ_CONTACTS" />

Step 2 - Add the following code in your activity asking for permissions at runtime.

if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.M) {
            if ((ActivityCompat.checkSelfPermission(this, Manifest.permission.GET_ACCOUNTS) == PackageManager.PERMISSION_GRANTED) && (ActivityCompat.checkSelfPermission(this, Manifest.permission.READ_CONTACTS) == PackageManager.PERMISSION_GRANTED)) {
                getGoogleAccounts();
            }

            else {
                ActivityCompat.requestPermissions(this, new String[]{Manifest.permission.GET_ACCOUNTS, Manifest.permission.READ_CONTACTS}, 1);
                //return false;
            }
        }

Step 3 - Add code for onRequestPermissionsResult -

 @Override
    public void onRequestPermissionsResult(int requestCode, @NonNull String[] permissions, @NonNull int[] grantResults) {
        super.onRequestPermissionsResult(requestCode, permissions, grantResults);
        if(grantResults.length > 0 && grantResults[0] == PackageManager.PERMISSION_GRANTED){

            getGoogleAccounts();
        }
    }

Step 4 - Finally add code to retrieve account using AccountManager

private void getGoogleAccounts(){

   AccountManager am = AccountManager.get(this); // "this" references the current Context
    Account[] accounts = am.getAccountsByType("com.google");

    for (Account acc : accounts){
        System.out.println("http accounts " + acc);
    }
}

Please refer to following link for changes in android 8 - https://developer.android.com/about/versions/oreo/android-8.0-changes#aaad