What happens to my email when I change web hosts?

Nothing, if the move is done carefully - and everything, if the new host rewrites your DNS wholesale. This is one of the most common self-inflicted mail outages:

• Web and mail are different records - the A record points at the website, the MX records point at the mail service. They are frequently different providers.
• A new host often supplies a full DNS zone - and their default MX records point at their own mail, not yours.
• Record your MX, SPF, DKIM and DMARC first - before anything is changed, so you can restore them exactly.

Check mail delivery in both directions immediately after the change, not the next morning.