I've seen dozens of articles on this subject over the past few years while doing research for various projects, so I have become pretty familiar with the subject of IPv4 and IPv6. That being said, I have read numerous articles that were pages upon pages long, which ended up summarizing to the same few points to answer the question of "what is IPv6?" Rather than write a ridiculously long article on the subject, how about the Cliff Notes version?
IPv6 is an acronym for Internet Protocol Version 6, and is the successor to IPv4. That being said, IPv6 is also referred to ad IPng (Internet Protocol Next Generation). The Internet Protocol is a packet-switching networking protocol created by the Internet Engineering Task Force (IETF) that is used to assign addresses to devices connected to a network, so other devices will know how to find them. This address serves the exact same purpose as a postal mailing address; it allows someone to send letters (or datagrams in this case) from point A to point B. IPv6 was primarily created to address an issue with IPv4, in which the world just happened to run out of assignable IP addresses!
IPv6 addresses the IPv4 address shortage by changing the length of addresses from 32 bits to 128 bits, effectively adding an additional 2^96 addresses (about 7.9 octillion, for a total of nearly 3.4 undecillion allowable addresses; 3.4*10^38). That's quite a gain compared to the 4.3 billion that IPv4 allowed for. While they were at it, the IETF also bolstered security (or at least planned for it), simplified the packet header format, added better support for multicast and Quality of Service (QoS) routing, and added support for an auto-configuration mode, which will allow devices to create their own address without user intervention. These changes have also allowed the concept of NAT (Network Address Translation) to no longer be needed. This is very handy when it comes to Internet of Things (IoT) connected and mobile devices!
Well, yes and no. IPv6 and IPv4 can run in a dual-stack environment, meaning that your old devices will continue to work, and will be able to communicate with IPv6 devices (using many other protocols that allow the gaps to be bridged). Some of your devices probably already support IPv6. Should you know if they support it? Not unless your a Network Admin or Software Developer. If not, then do your devices still work? If so, don't worry about it. If not, replace them; the replacement devices will, most likely, support IPv6.