This year on June 30th at 23:59:59 UTC, rather than the 59th second ticking to 00, there is an additional second added so clocks will tick to 60 before going to 00.
If this sounds rather odd, check out a more detailed explanation here on Wikipedia.
So what does this mean for us as software developers? Well, for Java JVMs, you can use the provided tzupdater tool (updates described here, and usage described here), download the updated timezone file and apply the update. For other platforms and operating systems, check release notes and updates from your vendors (e.g. For RHEL, check here).
Should you be worried about the impact to your systems? Well that depends on how sensitive your system is to the date/time and synchronization with other systems that you integrate with. For example, the ICE futures exchange have announced they are taking their systems down for an hour overlapping with 23:59:59 on June 30th to avoid any unexpected issues. Whether you have to go that far depends on your system.