Hi SuperKeet.
Are you working with a POP or an IMAP account? For POP accounts, the email may only appear on the device that you first used to open it. If this is an IMAP email account, check the settings that you have for Mail notifications. The resource below explains how to use this feature. Try turning off the notifications for Mail. When you turn them back on, try setting the notification to a banner vs. an alert that requires action.
Get help if you have issues using a POP email account with more than one device
If you use the same POP email account on more than one device, email messages might appear only on the first device that gets them. This happens when an email app is set up to automatically remove messages from the server after it receives them.
Mac Basics: Notifications keep you informed
Notifications work with your apps to let you know when something has changed. For example, you can see when there's incoming email in Mail, or upcoming appointments in Calendar. Each app displays these notifications as a banner, alert, or badge.
Banners appear in the upper-right corner of your screen, without interrupting what you're doing. Banners appear for a short period of time, then slide off screen to the right.

Alerts are important announcements that appear similar to banners. Alerts remain on your screen until you take action. Many alerts include buttons you can use to interact with them without having to open the app that generated them.
To dismiss an alert notification without interacting with it, use your pointer to swipe the banner to the right.
With Notification Center, you can see all of your notifications at a glance. To open Notification Center, click its icon
in the menu bar, or swipe left with two fingers from the right edge of your trackpad.

All of your recent notifications are listed in the Notifications tab. You can quickly respond to outstanding messages or emails directly from the Notifications tab. You can also click a notification to open its related app.
You can change how apps notify you from the Notifications pane in System Preferences. Choose System Preferences from the Apple menu, then click the Notifications icon. For more information about Notifications, click the Help button
in the Notifications preferences window.
Take care,