Q: Chrome not updated - what does this mean?

The message means you are running a very old version of Apple's OS X operating system.  OS X 10.6 (Snow Leopard) was released in 2009.

Google Chrome is no longer being updated with changes that can be used with such an old Apple operating system.

You either should purchase a new computer or update OS X if you can.

It means exactly what it says. You would need to upgrade to Mavericks or above for continued upgrades to Chrome software.

There is a big difference between an update and an upgrade.

Updates are new software versions that typically can be used with your current operating system version.

Upgrades are major version changes, often requiring a fee, that may well include operating system major changes.  For example, moving from Snow Leopard to the next OS X version (Lion) actually was an upgrade.

So you would not have been informed that an upgrade was available.  And no updates were available because you had the latest version of Snow Leopard.

If you are running Snow Leopard, as your Tag indicates, there are no updates for it beyond 10.6.8. It's a six year old version of OS X, the last update was 5 years ago.

You're confusing "Updates," with "Upgrades."  The current version of OS X is El Capitan (10.11.3). Which means that it has been updated 3 times since its release last Fall.

The message from Google means that they are not going to update Chrome for OS X versions that are more than 3 years old.

Your Mac meets the requirements for El Capitan. It can be obtained by using the App Store app in your Applications folder. You should make a backup of your current system to an external hard drive, preferablty a clone backup, before upgrading, as upgrades to an OS can't be uninstalled, only erased and replaced.

Perhaps you can run El Capitan.  Check here:

Upgrade to OS X El Capitan - Apple Support

"Will the hardware be damaged if you don't upgrade?"  Well, no.  "Will it be really bad ... if I don't do an upgrade?"  Well, yes.

But there's a probability model at work there, just as there is if you drive without an oil change.  As time progresses, you increase the probability of a catastrophe.

You are not endangering your hardware, but right now, your operating system is simply outdated so much that Google's Chrome (and undoubtedly other products) can no longer run on it.

One example would be if you had an iPhone or iPad, you could not use iTunes to communicate (e.g., synchronize, restore) between these devices and your computer because your operating system is too old to support the current iTunes.

So you can avoid all updates, and bunch a lot of changes together when one day in the future you must do a massive upgrade, or instead you can do frequent small updates and stay current on all your software.  I would always prefer the latter approach, given the reliability of Apple software when performing these frequent small updates.