An MX record, short for mail exchanger, is a DNS record that tells the rest of the internet which servers accept email for your domain. When someone emails you, their mail server looks up your MX records and delivers to the servers they name. Change the MX records and your email goes somewhere else.
How it is used
A domain usually has one or more MX records, each with a priority number. Sending servers try the lowest number first and fall back to the others if it does not answer.
The values identify the email provider. Records ending in mail.protection.outlook.com mean Microsoft 365. Records at google.com mean Google Workspace. Other values usually point at the registrar's or web host's own email service.
Why it matters to a business
MX records are the reason a website can move without touching email. The website is found through A and CNAME records; email is delivered through MX records. Leave the MX records alone and email carries on regardless of where the website lives.
They usually travel with SPF, DKIM and DMARC records, which prove that email sent from your domain is genuine. Those belong to email too, and should be left alone in a website move. Can I move my website without moving my email? has the full list.