Yes: the current IPv4 based Internet will continue to work
Currently the Internet uses IPv4 IP addresses, and you can reach all Internet destinations (websites) over IPv4. As long as both yourself and everybody you would like to communicate with (end-users, websites) have an official IPv4 IP address, the Internet will work and continue to work (for you).

Existing IPv4 addresses will continue to work for many years
The depletion of the pools of free IPv4 addresses (the IANA pool now and the RIR pools in the future) will have no impact on existing IPv4 addresses. At some point in the future IPv6 will become so widespread that people will start to turn off IPv4 and the usability of IPv4 will start to decline, but we can't even speculate about when this will happen, certainly not within the next five years.

New users will have a problem in the future
The problem with IPv4 exhaustion is primarily for new parties which don't have IP addresses yet, and parties which don't have enough IP address and need more. This is not a problem yet: currently IPv4 addresses can still be obtained without a problem. But at some point in time in the future, this will no longer be possible or become extremely difficult. We expect it will become difficult or impossible to get large blocks of IPv4 addresses from the RIPE NCC between the end of 2011 and mid-2012, but small blocks, especially blocks necessary to allow IPv6-to-IPv4 gateways, will likely be available for several more years.

However, just ONE IPv4 address might be enough for you!
For content, on one single IP address it is possible to host many, many websites, as websites can share the same IP address.
For access, behind one single IP address one can hide many, many end-users using NAT (Network Address Translation).
So for many applications a very small amount of IPv4 addresses would strictly speaking be enough to get and/or keep essential IPv4 based Internet services in operation. So as long as there are providers which still have, even only a few, IPv4 IP addresses available, it will still be possible to operate on the IPv4 Internet as we know it. This might continue to be possible for decades.

Or at least 256 then...
If you want to be independent of any provider and do your own routing on the Internet, you will need a block of 256 IP addresses (called a /24 or historically a Class C). The Internet will not route smaller address blocks then this. This is still perfectly doable now, but will become more and more difficult over time.

At IANA: yes, now
IANA is the organisation that maintains the global pool of unallocated IPv4 space. This pool has now run out.

At RIPE: no, not for another year
RIPE is the organisation that maintains the address pool for the (greater-)European region and is called a RIR or "Regional Internet Registry". RIPE distributes (allocates, assigns) IP space in the European region. It's pool of IPv4 address space for this was, until now, replenished by IANA when needed. With the exhaustion at IANA, this pool will not be refilled anymore. However, estimates are that RIPE still has enough IP space in it's pool to last for another year under the current usage rate and under the current allocation policies.

At LIRs: no, between 1 and many years
Local Internet Registries or LIR's, often Internet Providers, do part of RIPE's job, assigning IP addresses to their customers from an allocation (reservation) they have received from RIPE. When currently their IPv4 allocation runs out, they receive a new allocation from RIPE. Once the RIPE pool has exhausted, they will not be able to receive a new allocation anymore. Currently all LIRs still have space available in their allocation, and they will be replenished as long as RIPE still has space. When the RIPE pool has run dry, there will be a huge distribution between how long LIRs can still last with their allocation. Some will run out shortly after, while others might even be able to assign space for years or even decades.

End-users: no, between 1 year and never
End-users receive address space from LIRs. If end-users need more IPv4 IP addresses, they should normally be able to get them from their LIR for at least a year from now, and depending on the LIR they use maybe even for many years from now.

1 year from now?
Under the current allocation rate at current procedures, the RIPE pool should last for about another year.

Shorter?
As everybody gets more aware of IPv4 space running out, people might make a 'run' on IPv4 space at RIPE in the next few months, causing the allocation rate to rapidly rise, and making the RIPE pool run out much faster.

Longer?
RIPE is rapidly tightening procedures and justification requirements for registering IPv4 space. Where previously it was possible to request IP space for your needs in the next two years, now you can only register the amount of IP addresses you need in the next 6 months. Per July 1, 2011, you can only get the amount of IP addresses you need for the next 3 months. And RIPE demands more and more proof to justify your request. So it is already rapidly becoming more difficult to get IPv4 space registered. This might slow the allocation rate down, and might make the RIPE pool run out much slower.

Pareto Principle
Quoting Iljitsch van Beijnum of BGP Expert.com for the below text: 'With respect to running out of addresses, it's important to realize that the Pareto principle (the 80/20 rule) applies: out of the 7686 address blocks given out last year, only 392 (5 percent) were blocks larger than 100,000 addresses, but those were responsible for 82 percent of the address space given out. Even when the RIRs are no longer able to give out those large blocks, they may still be able to fulfill the requests for address blocks smaller than 10,000 addresses. Last year, 6425 such blocks were given out, totaling 14.03 million addresses. It really only takes a single address to be in the content business; it's the ISPs that need a continuous supply of new addresses to connect new customers. So the address shortages looming beyond the summer will hit ISPs and their broadband/mobile customers first and foremost, and the content industry to a much lesser degree.'

Request PI space
Provider Independent or PI space, so space you can take with you while moving to a different LIR or different provider, can be registered with existing LIRs for as long as RIPE still has space available in it's pool. See previous item for estimations on how long that is going to be. Do notice that it is not possible to sub-allocate PI IP's you have received to your customers.

Request PA space
Provider Aggregatable or PA space, so space you need to use specifically with services of that provider, can be registered with existing LIRs for as long as they have unassigned allocation space available. Some will run out of space shortly after RIPE runs out, others will last much longer. Again, notice that it is not allowed to sub-allocate PA IP's you have received from your LIR, to your customers.

Become LIR yourself
If you become LIR yourself, and you can justify your IPv4 address space need, RIPE will allocate a block of IPv4 space for you as reservation to make an initial and future assignments to yourself and your customers from. So as a LIR you can get IP space for yourself and your customers (as long as you and/or they can justify their need according to the RIPE rules). Becoming LIR is expensive (2.000 Euro initially, and at least 1.300 Euro yearly) and requires a lot of reading, education, etc.
RIPE has however adopted a policy for the last bit of space it has available, so even when their normal pool runs out, it will still be possible to get IPv4 space for and establish new LIRs (for some time):

5.6 Use of last /8 for PA Allocations
The following policies come into effect as soon as RIPE NCC is
required to make allocations from the final /8 it receives from
the IANA. From then on the distribution of IPv4 address space
will only be done as follows:
   1. Allocations for LIRs from the last /8
      On application for IPv4 resources LIRs will receive IPv4
      addresses according to the following:
      a. LIRs may only receive one allocation from this /8. The
         size of the allocation made under this policy will be
	 exactly one /22.
      b. LIRs receive only one /22, even if their needs justify
         a larger allocation.
      c. LIRs may apply for and receive this allocation once
         they meet the criteria to receive IPv4 address space
	 according to the allocation policy in effect in the
	 RIPE NCC service region at the time of application.
      d. Allocations will only be made to LIRs if they have
         already received an IPv6 allocation from an upstream
	 LIR or the RIPE NCC.
						

Managed LIR service
Some parties, like NL-ix, offer a Managed LIR service. This means you formally become RIPE LIR yourself, but you let for example NL-ix manage all practicalities of it.

'Buy' IPv4 space
IPv4 addresses are a public resource and as such cannot be sold or bought. However, if you buy a company or network which currently has assigned IPv4 addresses, you are allowed to continue to use those addresses, as long as the original criteria under which the IP space was assigned by the RIPE LIR are still valid. RIPE will want to validate if this is still the case and you will need to provide proper justification. But if you can provide such a justification, and are able to find a network which currently has IPv4 addresses and is for sale, this is a solution which can work for years and even decades from now.

What is IPv6?
IPv6 is the replacement of IPv4, and supports extremely much more addresses then IPv4. IPv6 has been developed more then a decade ago, and it is by now supported by just about every operating system and router that has been brought on the market since.

Is IPv6 actually used?
Yes!
Although the IPv6 Internet is still very small, about 1/75th of the IPv4 Internet in terms of routed networks, it is used on the Internet backbone and it's use is rapidly growing.

Are the IPv4 and the IPv6 Internet interconnected?
No!
Users of the IPv6 Internet can reach the entire IPv4 Internet via transition mechanisms (like SIIT, 6to4), but the other way around IPv4 Internet users cannot reach IPv6 users, as they can simply not address all IPv6 destinations in an IPv4 format packet (other then via a workaround like DNS64).

Is using only IPv6 (and no IPv4) an option?
Using only an IPv6 IP address for your service would make the service reachable only for IPv6 users and unreachable for IPv4 users. Therefore, everybody providing a service on Internet, even if they primarily use IPv6 for the service, will for as long as possible try to also give their service an IPv4 address, so it is still reachable for the 'old' Internet.

When do I really need to have IPv6?
The sooner the better!
But it depends very much on your type of business. For instance, pure web businesses don't need IPv6 as much, because HTTP is easy to proxy or translate. ISPs that want continue to be able to connect new customers will probably have to deploy "carrier grade NAT" or something similar in their networks so that multiple subscribers can share an IPv4 address. However, this breaks many more protocols than a regular NAT box at the customer, so it's important to also provide customers with IPv6 access so they can run protocols that don't work through the carrier grade NAT system over IPv6 instead.

How can I deploy IPv6?
RIPE has some information about IPv6 deployment, and you can always contact your current provider or NL-ix for more info.

NL-ix IPv6 course
NL-ix offers a IPv6 course 4 times a year.

Book: Running IPv6
The book Running IPv6 provides good insight about how IPv6 works.

BGP Expert
Check out the BGP Expert.com website of Iljitsch van Beijnum for more background articles on this subject.

Geoff Huston
Geoff maintains the IPv4 Address Report in which the date for exhaustion of the IANA Unallocated Address Pool was predicted. This document gives a lot of background information around the depletion.

ARS Technica: Everything you need to know about IPv4
ARS Technica has a nice article about IPv6.

RIPE
RIPE is the Regional Internet Registry (RIR) for the (greater-) European region. RIPE has a great FAQ about IPv4 depletion on their site.

IANA - Internet Assigned Numbers Authority
Go to the IANA website for more information about what the role of IANA is in allocation of number resources like IPv4 addresses.

ISOC - Internet Society
ISOC has a great FAQ online about IPv4 depletion and IPv4.