When most people hear "flag" concerning email, they mistakenly think that means "mark as spam" all the time, and that’s incorrect.
Email flagging has been around more or less for as long as email has existed. It’s a special mark you can place on any message so that you can quickly retrieve it later.
Small example: An email comes in that’s a notice for a bill you have to pay, and you decide that you’ll pay it tomorrow. The problem however is that mail is swamped in with a whole bunch of other emails. While true you could just drag it to a "Bills" folder or search for it later, why do that when you can simply flag the message and find it quickly that way?
The ability to flag messages is to the best of my knowledge available in every mail client and just about every webmail system.
"Is the flag the same as email importance?"
No. Email has the ability to have ‘importance’ set on it (as in low, medium or high priority), but that’s not what the flag is.
Setting an email flag in…
Microsoft Outlook Express 6
This is an ancient email client but it does have flagging ability. Click View > Columns, enable Flag, and you’ll see the Flag column it in your message list. Click the white space in the flag column next to any message and a small red flag will appear. Click again and it goes away.
And yes it will work with POP or IMAP.

Mozilla Thunderbird 9.0.1
The flag column is called Star in Thunderbird and is enabled by default.
Like with OE6, this will work with POP or IMAP.
Hotmail
The flag column is in the center of your message list. Flags are grayed-out by default until you click them, and then they’re enabled.

Yahoo! Mail
The Yahoo! system calls their flagging "starring", and is available at the far right side when viewing your message list.

Gmail
In Gmail, flagging is also "starring" and is on the left side of your message list.

What’s the main advantage of flagging emails?
Flagging is without question the absolute easiest way to retrieve important emails later.
You see an email come in that you know you’ll want to go back to later, so you flag it. At that point you don’t have to remember who it’s from or even what it’s about. All you have to do is search or sort by flagged messages when you want to see your ‘important mail’ list, which all mail clients and webmail have as a feature. When finished, just unflag the messages you’re done with.
While true this is an all-or-nothing way of sorting mail (meaning "this is important" or "this isn’t important"), that’s how most people treat email anyway, and using flagging is much easier than messing around with folders (or ‘labels’ in Gmail), filters, etc. Email is supposed to be easy, and the flag feature is exactly just that.

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