According to Apple, its latest Macs are powered by 64-bit processors, and this allows 64-bit apps to access more memory and give faster system performance.
When you download macOS 10.13.4 and open a 32-bit app, you'll receive a notification that tells you the software is "not optimized" for the Mac. "They say "[app name] is not optimized for Mac" and they only appear once so as not to be a pest, but it's a gentle nudge for users and developers alike that the future for Apple is very much going to be based on 64-bits. To help explain what is going on, the message features a Learn More button. However, when the successor to High Sierra begins rolling out later this year, 32-bit apps will no longer run and eventually, they may even be wiped from the Mac App Store.
Apple has launched a campaign to inform users of the imminent cessation of support for 32-bit programs.
Apple performed a similar transition with its iOS 11 software, warning users before cutting support for 32-bit applications entirely. Still, when the big day finally rolls around, there will nearly certainly be complaints nonetheless.
Presumably, the switch will officially be made in macOS 10.14, but Apple maintains that "final transition dates have not yet been established".
As a reminder, new apps submitted to the Mac App Store must support 64-bit, and starting June 2018, app updates and existing apps must support 64-bit.
If you have a favorite app that hasn't been updated in a while, it might be worth contacting the developer to see if they're planning on issuing an update. Click it, then choose "About this Mac".
Apple's transition from 32-bit to 64-bit actually started 15 years ago with the introduction of Power Mac G5. Developers might still have a few months to work on their apps and transition into 64-bit, if they haven't already. When you select an individual application, you will see a field titled "64-bit (Intel)". Presently, the App Store does not accept submissions of 32-bit apps for macOS.
Back in January, Apple's beta release notes for macOS 10.14.3 said that these notifications would begin happening with that version's public release, but they didn't start when that update first became available.
This break with compatibility for the very first version of watchOS is an indication that Apple intends to keep rushing forward with Apple Watch, and that watchOS 5 will be more than just a minor upgrade.