Receive URL in my App, when you click share in the Browser

  

How can I make my application appear when sharing?

What you want is something similar to an Intent Chooser

for this you have to add a Intent-Filter to the main Activity of your application for Receive an Intent of the type ACTION_SEND :

<activity android:name="ShareActivity">
    <intent-filter>
        <action android:name="android.intent.action.SEND"/>
        <category android:name="android.intent.category.DEFAULT"/>
        <data android:mimeType="text/plain"/>
    </intent-filter>
</activity>

You can see this in the documentation Reception of an implicit intent , for example by defining this in MainActivity :

    <activity android:name=".MainActivity">
        <intent-filter>
            <action android:name="android.intent.action.MAIN" />

            <category android:name="android.intent.category.LAUNCHER" />
        </intent-filter>

        <intent-filter>
            <action android:name="android.intent.action.SEND" />
            <category android:name="android.intent.category.DEFAULT" />
            <data android:mimeType="text/plain" />
        </intent-filter>

    </activity>

In this way, when you share, you can see your application as an option to do it:

  

How can I receive the shared url, in a field of my app?

To receive data that you previously sent using a Intent , these are received in the onCreate() method using the getIntent()

 Intent intent = getIntent(); 
String urlCompartido = intent.getStringExtra(Intent.EXTRA_TEXT);

or simply:

 String urlCompartido  = getIntent().getStringExtra(Intent.EXTRA_TEXT);

Try specifying an activity that will receive mimeType="text/plain" in the manifest:

<activity android:name=".ActividadLink">
    <intent-filter>
        <action android:name="android.intent.action.SEND" />
        <category android:name="android.intent.category.DEFAULT" />
        <data android:mimeType="text/plain" />
    </intent-filter>
</activity>

This means that the ActividadLink will be the activity that will be executed when a link is shared. Then to get the link it would be this way:

if (Intent.ACTION_SEND.equals(action) && type != null) {
    if ("text/plain".equals(type)) {
        String link = intent.getStringExtra(Intent.EXTRA_TEXT);
            if (link!= null) {

                textBox.setText(link);
            }
    }
}