Check Settings -> Notifications -> Reminders.
Make sure that "Allow Notifications" is turned on and that there is a notification sound set.
Also make sure that the alert style is either "Banners" or "Alerts".
Also ensure that you're actually setting your reminders up with notifications.
Tap the little 'i' icon when you're entering a reminder and make sure to choose "Remind me on a day" and choose a date for the reminder.