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Changing a mailbox password is a single action in the control panel. The part that catches people is everything else that was using the old one, which does not know it has changed.
Where the password lives
A mailbox password belongs to the mailbox rather than to your account, which is the distinction that matters when you are looking for it. It is set on the mail service in the toolbox, not on the account sign-in screen.
That separation is deliberate. The account password gets you into the toolbox to manage billing and services. A mailbox password gets one address to read its mail, and a person can be given a mailbox without being given the account.
Changing it
Open the mailbox in the mail service and set a new password. The change takes effect immediately for new connections, and the mail already in the mailbox is untouched.
What happens next is the part worth planning for. Every device and client that was signed in with the old password is now holding a credential that no longer works, and they fail at different times depending on how often they check.
Everything that needs updating
Think of it as a list rather than a single device. Every phone and tablet with the account on it. Every desktop client, which is often more than one on the same computer. Any device with mail set up as a secondary account rather than the primary one, which is the one people forget. And any application password that was issued for a specific device or app.
Application passwords are the piece that surprises people. They are separate credentials generated for one device, and they do not change when the mailbox password changes. If a phone was set up with one, changing the main password will not fix it and will not break it either. If you want that device to stop working, revoke its application password rather than changing the mailbox password.
Most clients reconnect on their own once the new password is entered, and no mail is lost in the meantime because it is held on the server. A device that has not been updated simply stops syncing, and setting up mail on your phone walks through reconnecting one from scratch if editing the stored password does not take.
If a phone keeps asking
A device that repeatedly prompts for a password after you have entered the new one is usually one of three things.
The most common is a second account on the same device still holding the old password, which prompts under the same mailbox name and looks like the one you just fixed. The second is an old entry stored in the device's password manager rather than in the mail app itself, which overwrites whatever you type. The third is an application password that was revoked.
Removing the account entirely and adding it again clears all three, and it is usually faster than working out which one it was.
If somebody has left
Changing a password is not the right tool for removing access, because it leaves the mailbox and its mail intact and simply locks a person out of reading it. If the mailbox belongs to the business and the person has left, the mailbox itself is what to deal with, and the mail in it is business correspondence rather than personal property.
For your own account rather than a mailbox, logging in to your account covers the account side, including what to do when the sign-in itself is the problem.