Email on my iPhone won't work like the one on my MacBook Air

Thanks, everyone I finally figured it out. YAY! What I did was:

1. On my iPhone, I went to settings.
2. I select email
3. I scrolled to email accounts and selected the Gmail account (which I had already deleted on Google)
4. then at the bottom of the screen I selected (delete account)
5. Then a screen came up that asked me if I REALLY meant it, and I hit "delete" again. Then it was GONE!

I wasn't clear enough. Sorry. I'm not talking about my Apple account. I have had an Apple account since the green screen days. I have had an Apple iCloud email address for many years, and I would never want to change it in any way. I'm talking about my email clients on both my MacBook and my iPhone. When I got a Gmail address a few years ago, I had both mailboxes (the Gmail and the iCloud) downloaded to my email client on my MacBook and on my iPhone. But I quit the Gmail address as I wasn't using it much. I went to Google and deleted it from their Gmail servers. Then I went to my email client on my MacBook and went to the settings and deleted Gmail boxes from my MacBook email client. I assumed that it would disappear from my iPhone, too, because my iPhone and MacBook do a lot of stuff in common via iCloud, etc. But when I went to my phone just now, the Gmail folders were still on the client, and I don't know how to delete them. I'm hoping someone will help. This Community is loaded with geniuses who are far better at Mac and iPhone stuff than I am. :)

@ckuan - As to what third-party address I had then, I have no Idea. It was more than 30 years ago. iCloud didn't even exist yet when I got my first Apple 2 computer, nor did Macs. Everything was very basic. I had a website and probably had a dot net address by the time iCloud came along. I only got the Gmail address a couple of years ago, when I got a paid YouTube account to get rid of the ads on my AppleTV. And then Google started getting up in my business, so I dumped the Gmail address, and couldn't get their mailbox off my iPhone. Every time I tried to check my email on my phone, Google would send me a message to come back. It was really creepy! I guess, whenever I went to the mail client on my iPhone, it triggered a download command to the old account on Google that no longer existed. I'm glad I finally figured it out. Thanks for your help.

You cannot delete your Apple ID (Apple Account)

you don't have to use but your Gmail is your Apple Account.

Again start slowly.

When you first got your @iCloud.com address from Apple

what 3rd party email address were you using to sign into Apple to get it?