Of course, we have a currently limited amount of IP addresses that can be given out. When IANA/RIRs were formed, IANA gave allocations of /8's (the largest groups of IPv4 addresses) out to all of the RIRs, who then assigned them to organizations within their respective areas. (For example, APNIC - Asia Pacific, AFRINIC - Africa, ARIN - North America) The problem now is that we are simply running out of addresses with the current number of internet connected devices.
Now, we resolve this within our own homes by using routers that support NAT (Network Address Translation) which is fine for our own local sites. However, a large amount of ISPs want to enable something called CGN (Carrier-grade NAT) which will take what happens on your own local router and blanket it over their entire network. This is where a large group of customers (including you) would have one single (or a handful) of IPv4 addresses that represent you on the public internet, but each site/home/office would have its own internal IP address to that ISP's network. Most cell phone providers actually already do this (although I'm not sure if we can call it CGN) where your phone has a Class C IP address assigned to it when you connect to the network, but all traffic goes through a single IP address.
Now there are problems to this:
1. It is hard to track who is doing what. If a customer of an ISP performs some sort of illegal action (bomb threats, hacking, etc use your imagination here. ;) ) It is hard to track down who is exactly doing what without your ISP simply logging all requests that you make. Instead in the current form, your ISP can be simply asked "Hey, who had this IPv4 address at this time?" instead of "Hey, what sites was this person looking at?"
2. Let's say that you were a member of some sort of forum or new site (take HN or reddit for example) and someone sharing your IPv4 address did something that caused that IP address to be banned. Now, you cannot access the site. This allows your access to the internet to be dependent upon whoever else is using that same address. Of course, this can (and has happened) already by sites blocking entire subnets (a lot of IRC networks do this for overseas networks that usually harbor abuse.)
3. It simply becomes confusing security-wise. Take into consideration that you had a VPS or server with a provider like Digital Ocean or Linode. You set up a firewall on that server to allow only your IP address in. Now, you've allowed everyone else sharing that same IP address to access your server if they knew the credentials.
With IPv6, we have more addresses than there are grains of sand on Earth (as the saying goes.) We also said that we'd never run out of IP addresses with IPv4, but the growth of the Internet was seriously underestimated!
Some ISPs will run dual-stack (I believe Comcast does now) where all users now have IPv4 and IPv6 addresses!