The KiwiSDR is a US$299 HF SDR that can monitor the entire 0 - 30 MHz band at once. It is designed to be web-based and shared, meaning that the KiwiSDR owner, or anyone that they've given access to can tune and listen to it via a web browser over the internet.
OpenWebRX is code originally created by András Retzler and a modified version runs on the KiwiSDR devices. This code is what allows them to be accessed online by a browser and was popularized by it's use in the KiwiSDR. The original code can also be used by other compatible SDRs such as the RTL-SDR.
Recently András released news that he is discontinuing work on OpenWebRX due to interest in other projects, but it will remain on GitHub as open source code. András also notes that the security of OpenWebRX will soon be in question as it utilizes Python 2, which has been designated end of life on January 1 2020. In addition, if you've been following OpenWebRX since the beginning, you'll know that in the past OpenWebRX was involved in an legal/ethical issue over open source licencing with KiwiSDR. Although the problems with KiwiSDR were resolved amicably, Andras also references his frustrations with similar situations to do with his code being forked again and again.
We note that maintenance and development of the KiwiSDR OpenWebRX code will continue as they are considered separate projects. Due to some confusion, we importantly reiterate that the KiwiSDR product is unaffected by OpenWebRX being discontinued. Although KiwiSDR is based on OpenWebRX they use their own custom branch of the software that is maintained by the KiwiSDR owners and not by András.
András also runs the popular sdr.hu OpenWebRX/KiwiSDR directory, which was/is considered the main directory for finding and accessing public KiwiSDR and other SDR devices running OpenWebRX. Recently the directory was restricted, and now can only be accessed by those with a ham radio callsign. It is unclear why this decision was made as sdr.hu was very popular with shortwave listeners and radio newbies who are typically not hams. But the sdr.hu FAQ notes "The purpose of the site is to serve amateur radio. I decided to restrict access to the receiver list in order to protect the site and its purpose in the long term."
Fortunately, over on his SWLing.com blog, Thomas has noted that there are still other KiwiSDR directories available such as https://ve3sun.com/KiwiSDR, http://kiwisdr.com/public and http://rx.linkfanel.net.