I get a lot of email, and sometimes I miss something, or I don't notice something urgent right away.
Here's how to configure Gmail on your Android phone to alert you distinctively when you get an urgent email.
First, identify what is important. It might be emails from someone in particular, or with a particular topic, etc. Most recently, I configured a special alert for emails from a particular notification service that always have a certain string in the subject.
Then, in the Gmail web UI, set up a filter for these messages in Gmail and assign them a new label which is specific to this class of messages.
Now, from your phone, go to that label, and from the menu icon in the upper right, choose "Label settings". Then, set this label to sync to your phone (for any duration), and you can then choose a notification sound. Choose a sound you don't use for other notifications on the phone.
Now, whenever you get an email from this category, you'll get a distinctive sound which tells you what's up without you having to even unlock the phone.
You may be able to use this ability to specifically notify on the emails you care about to reduce the alerting for emails you don't care about. You can turn off notifications for new messages in categories or labels that are not urgent. Then, when your phone goes “ding”, you'll know that it's something worth interrupting you for.