There is no lack of geeky explanations on the internet to explain what functions Exchange MAPI and ActiveSync perform. Let's make it simple for the average person.
Both MAPI and ActiveSync are Microsoft products and, generally speaking, work with some form of Exchange (Hosted Exchange or Microsoft 365 Exchange). Exchange is what hosts your mailbox "in the cloud".
It helps when you know a little history. Microsoft first released Exchange back in the mid 1990s. Back then, desktop computers were dominant (and to some degree laptops) and Microsoft created Exchange as the "Server" and Outlook as the "Client". They both are designed to work together.
MAPI - Is what syncs the Exchange server to Outlook that resides on a desktop or laptop computer. You know, Outlook email, email folders. Calendars, Contacts, Tasks, Notes, etc.
Not until handheld devices, mainly Smartphones, and Tablets, did ActiveSync really take off in the late first decade of the 2000s.
ActiveSync - This technology syncs your phone and tablet with your Hosted Exchange or Microsoft 365 Exchange email service.
This works because if you send an email or add a contact or set up a Calendar event on your phone, it is synced with the Exchange server. Later when you access your Outlook or phone, the input is showing in your other devices.