How do I update my Mac from High Sierra 10.13.6 to latest version?

The model year of an iMac is available at Apple menu > About This Mac.

A late 2009 through mid 2011 iMac can be upgraded to High Sierra 10.13.6 but no later version. 

A late 2012 through late 2013 iMac can be upgraded to Catalina 10.15.7 but no later. 

A mid 2014 and late 2014 iMac can be upgraded to the latest version of Big Sur macOS 11 but no later. 

A late 2015 or mid 2015 iMac can be upgraded to the latest version of Monterey macOS 12 but no later. 

A 2017 iMac can be upgraded to the latest version of Ventura macOS 13 but no later. 

For all of the above, see instructions in: How to download and install macOS - Apple Support

A 2019 or later iMac can be upgraded to the current version of Sonoma macOS 14. See: How to upgrade to macOS Sonoma - Apple Support

Be aware that if you upgrade to Catalina 10.15 or later you will also need to update any non-Apple 32-bit apps. See: https://www.stclairsoft.com/Go64/

or

 https://www.macworld.com/article/232735/how-to-check-if-mac-software-is-32-or-64-bit.html

macOS Mojave 10.14.6 - Technical Specifications

General Requirements

OS X 10.8 or later

2GB of memory

12.5GB of available storage (OS X El Capitan 10.11.5 or later)*

Some features require an Apple ID; terms apply.

Some features require a compatible Internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Mojave:

MacBook (Early 2015 or newer)

MacBook Air (Mid 2012 or newer)

MacBook Pro (Mid 2012 or newer)

Mac mini (Late 2012 or newer)

iMac (Late 2012 or newer)

iMac Pro (2017)

Mac Pro (Late 2013; Mid 2010 and Mid 2012 models with recommended Metal-capable graphics cards)

https://itunes.apple.com/us/app/macos-mojave/id1398502828?mt=12

Thanks to MartinR Terminal command…

softwareupdate --fetch-full-installer --full-installer-version 10.14.6

It will place a copy of the installer in your Applications folder and you can use that to create a bootable installer.

MacOS Catalina 10.15.7 - Technical Specifications

General Requirements

OS X 10.9 or later

4GB of memory

12.5GB of available storage (OS X El Capitan 10.11.5 or later)*

Some features require an Apple ID; terms apply.

Some features require a compatible internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Catalina:

MacBook (Early 2015 or newer)

MacBook Air (Mid 2012 or newer)

MacBook Pro (Mid 2012 or newer)

Mac mini (Late 2012 or newer)

iMac (Late 2012 or newer)

iMac Pro (2017)

Mac Pro (Late 2013 or newer)

How to download and install macOS - Apple Support

https://apps.apple.com/gb/app/macos-catalina/id1466841314?ls=1&mt=12

macappstores://apps.apple.com/app/macos-catalina/id1466841314?ls=1&mt=12

How to upgrade to macOS Catalina – Apple Support

And caution on 10.15 or higher, to see which Apps will no longer work in 10.15 get Go64...

https://www.stclairsoft.com/Go64/

macOS Big Sur - Technical Specifications

General Requirements

OS X 10.9 or later

4GB of memory

35.5GB available storage on macOS Sierra or later*

Some features require an Apple ID; terms apply.

Some features require a compatible internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Big Sur:

MacBook (2015 or later)

MacBook Air (2013 or later)

MacBook Pro (Late 2013 or later)

Mac mini (2014 or later)

iMac (2014 or later)

iMac Pro (2017 or later)

Mac Pro (2013 or later)

Learn about compatibility and how to identify your specific Mac model

https://apps.apple.com/us/app/macos-big-sur/id1526878132?ls=1&mt=12

And caution on 10.15 or higher, to see which Apps will no longer work in 10.15 get Go64...

https://www.stclairsoft.com/Go64/

A quick check of MacTracker confirmed this suspicion: If your iMac is old enough that it can run High Sierra – and it is not an iMac Pro – it is old enough that it can't run Sonoma (the latest version of macOS).

Whether you can upgrade to one of the intermediate releases of macOS will depend on the hardware model of your iMac (which you can find in About This Mac in the Apple () menu).

To update your Mac at the latest version check the Mac model and verify if your Mac is able to install a newer version of software. Then go to the App Store and search the MacOS version that you want and get it. Install the software and reboot your Mac.