The Internet has "growed like topsy" and done it very well - it's a wonderful example of a self-regulated collaborative "thing" that should be a model for us all. But the numerical addresses underlying domain names etc. are running out and, despite various fairly effective kludges designed to get around this, the IPv4 address space really will fill up.
So, we have IPv6, which (primarily) supports the vast number of addresses needed by mobile phones and "the Internet of things", each with its own address. However, it is also a refactoring of IPv4 so as to cope with commercial and personal Internet use simply not envisaged when the Internet was defined.
So, my first message is, that existing IPv4 addresses will continue to work, but your equipment must be aware of IPv6 if it might meet it and not simply crash with what it sees as an invalid address. This is really an IT governance issue - technology acquisition should ask about IPv6 support; and network (and, possibly, other) development should take a look at IPv6 features and innovation, to see if they can help to make systems design simpler, more reliable, or more secure. The value proposition for IPv6 is reduced complexity and greater reliability, resilience and security for business-critical Internet connectivity.
My second message is DON'T PANIC and, in particular, don't allow yourself to be exploited by vendors of technological snake oil, peddling "fear uncertainty and doubt" (or even by politicians, with an agenda, and infeasible plans for "legislating the Internet"). IPv4 will still work and natural replacement of technology means that your kit is probably IPv6-aware already (obviously, if Internet access is important to your business, you shouldn't just rely on this, however). Even if all your networking technology is antique and falls over if it meets an IPv6 address, you only need to replace outward-facing kit as a priority. There's a bit of a risk, I suppose, that packets with an IPv6 address arrive by an unanticipated route, so you should, at least, do an IPv6 risk analysis. Getting experienced external help with this is a good idea (but be careful of the technology equivalent of the PPI insurance claim companies that see you as a potential cash cow). Keep calm and carry on - and check out the official RIPE NCC (the Regional Internet Registry (RIR) for Europe, the Middle East and parts of Central Asia) "Act Now" site here - which gives reliable, independent and non-commercial advice...