If you are using an app to access your Office 365 account and it keeps asking you for your password it is usually because that app doesn't support 2-step verification. You'll need to either change apps, or use an app password. See more information in the "Setting up Office on your Mobile Device" section of the 2-step verification guide.
If it's an iOS device, try removing your account and re-adding it first. Apple's mail app needs to reconnect. If that doesn't work, your version of iOS may not support 2-step verification with the mail app.