In iOS, a "Local Notification" is used by an app when it is in the background, to inform the user that something has happened, that they might want to pay attention to:
Local notifications ... to inform users when new data becomes available for your app, even when your app is not running in the foreground. For example, a messaging app might let the user know when a new message has arrived, and a calendar app might inform the user of an upcoming appointment.
Apple dev - Local and Remote Notifications Overview
[Its "local" if the app itself is providing the new data; "remote" if a remote server is sending the update.]
Is there an equivalent on Android?
Use NotificationCompat.Builder if you are targeting old APIs too.
Intent intent = new Intent(ctx, HomeActivity.class);
PendingIntent contentIntent = PendingIntent.getActivity(ctx, 0, intent, PendingIntent.FLAG_UPDATE_CURRENT);
NotificationCompat.Builder b = new NotificationCompat.Builder(ctx);
b.setAutoCancel(true)
.setDefaults(Notification.DEFAULT_ALL)
.setWhen(System.currentTimeMillis())
.setSmallIcon(R.drawable.ic_launcher)
.setTicker("Hearty365")
.setContentTitle("Default notification")
.setContentText("Lorem ipsum dolor sit amet, consectetur adipiscing elit.")
.setDefaults(Notification.DEFAULT_LIGHTS| Notification.DEFAULT_SOUND)
.setContentIntent(contentIntent)
.setContentInfo("Info");
NotificationManager notificationManager = (NotificationManager) ctx.getSystemService(Context.NOTIFICATION_SERVICE);
notificationManager.notify(1, b.build());