Today, the majority of the internet (I'm going to guess some where around 99.8%) uses IPv4 internet.
When broken down, IPv4 has 32 bits of information.
Example: Your IP is 188.8.131.52 (not your REAL ip, this is an example)
In Binary, this is represented as 11001000.01111101.00110010.00000001
That right there is 32 bits of information. Each number (either a one or zero) is a bit.
This means, that IPv4 can support a total of 232 end points (4,294,967,296; but for networking reasons [broadcast and network addressing] the number is more accurately around 4.22billion).
That means there can be a total of about 4.2 billion machines on the internet at once. Now, that sounds like a lot, but when you take into account all the machines that are on the internet at all times (cellphones, servers, etcetc) you'll realize that 4.2 billion unique public IP addresses is not nearly enough.
So, in order to fix this "IPv4 Exhaustion", private IP addresses were issued (you've probably seen these, 192.x.x.x, 10.x.x.x, 172.x.x.x). With billions of machines running at all times, we're nearing exhaustion of IPv4 (estimated by 2011-2012) because we can only have 4.2 billion public end points online at a time.
Now, with IPv6, we will have a total of 128 bits for our IP.
Shall we compare these two numbers?
232 = 4,294,967,296
2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456
IPv6 is the long-term solution to our growing issue in IPv4.
IPv4 is still around today because of private IP's and NAT's. You can look those up or ask me about them if you want to know more.