The Importance of ProofOfConcepts
Any problem is an invitation to find a solution.
Any solution  at least in my line of work  is an amalgam of concrete implementations of abstract concepts. Each of those implementations may or may not meet the requirements just like any of those concepts may or may not be the right one for the situation at hand. You therefore need to prove two things:

the solution proposed is the solution to the right problem

the solution proposed solves the problem
The solution being a concrete implementation of an abstract concept, needs to be proven. Hence, you need a Proof of Concept.
Why? When? What? How? Who?
These are the five basic questions you need to have answered before being convinced of anything  at least, I need to have answers to these five basic questions before being convinced of anything and I try to have answers to each of these whenever I try to convince someone of anything. So, I will now try to answer each of these questions to convince you of the importance of proofs of concepts.
Why?
Ultimately, a proof of concept saves you money: it saves time because you don’t spend more time than necessary on a concept that you can’t prove; it saves more time because concepts that you can prove are known to work and can be built upon as the project progresses; it instills confidence in the solution in your management and development teams and it documents two things: what the solution is meant to solve, and how the solution is meant to solve it. It therefore also establishes two things  which are of vital importance to understand for any solution: the capabilities of the solution and the limitations of the solution.
I express functional requirements in terms of required capabilities or capacity and permitted limitations, and I believe this is a universal way of expressing functional requirements. Mind you that this does not express all types of requirements  e.g. security requirements can (and should) be expressed in terms of rights and obligations, but they are nonfunctional requirements (see L. Chung, B. Nixon, E. Yu, and J. Mylopoulos, NonFunctional Requirements in Software Engineering. Kluwer Academic, 2000; P. Devanbu and S. Stubblebine, “Software Engineering for Security: A Roadmap,” The Future of Software Eng., A. Finkelstein, ed., ACM Press, 2000.; and D.G. Firesmith, “Specifying Reusable Security Requirements,” J. Object Technology, vol. 3, no. 1, pp. 6175, Jan.Feb. 2004. and many others). A proof of concept proves beyond a reasonable doubt that the functional requirements that are to be met by the solution can be met by the solution proposed. Though it is not a complete implementation of the solution, it is enough of an implementation to prove the feasibility of the solution and that it fits the problem at hand. It should, however, be significantly less expensive to develop than the solution itself, lest it not serve its purpose as a time and money saver.
When?
“Before it’s too late, but no earlier”.
It’s too late to prove a concept when your design depends on the concept to work, so you have to prove it before that. It’s too early to prove a concept if you haven’t even analysed the requirements the solution is going to have to meet yeat, so you need to prove your concept after that. It’s no use to create a formal proof of concept if there are already plenty of proofs around, so you might not need to make your proof of concept at all (i.e. don’t try to prove the obvious).
“When in doubt, prove it!”
What?
Some things are not concepts and should not be treated as such; a thing does not become a concept just by sticking “conceptually” before it; and negatives cannot be proven.
How?
Say you want to use MySQL in a project written in C++, but you want the project to be closedsource and you don’t want to pay a license to MySQL AB (or Sun Microsystems, or (soon) Oracle). You don’t mind distributing MySQL’s own source code under GPL, but you don’t want to GPL yours. Think about this for a bit.
After thinking about it for a bit, you may have come up with the solution “we need an abstraction layer for MySQL that allows us to talk to the MySQL database without using MySQL’s own code”. You mull on that a bit, think about what that abstraction layer should be like and come up with something like this: “We need a solution in which a closedsource object, can load an opensource object at runtime; both expose the same or a similar API allowing to perform database queries and the client application can use that API, talking directly only to the closedsource part of the solution, to perform queries. The closedsource object shall not depend on the open source object in any way shape or form, but the open source object may depend on the closed source object”. Call the closedsource object “Manager” and the opensource object “Driver” and you get ODBC.
You now have the following assertions:

“ODBC allows us to perform SQL queries from within the C++ code on a MySQL database”

“ODBC meets the performance requirements for our solution”

“Using ODBC allows us to use the MySQL database without rendering our own source code GPL and without paying for a license” Each of these assertions is actually a hypothesis and each of these hypotheses can be tested.. In order for a hypothesis to be tested, you need to be unable to falsify it  i.e. you need to try and fail at falsifying it; and you need to prove it practically feasible.
Our first assertions, “ODBC allows us to perform SQL queries from within the C++ code on a MySQL database” is one that can only be falsified by trying to prove its feasibility and failing. I’ve tried it  it’s feasible (it’s actually very straightforward). The second assertion, “ODBC meets the performance requirements for our solution” depends on our performance requirements. Once you have those, you can try to falsify it by building on your first proof of concept  the one that proved that you can perform a query from C++  by performing queries that you conceive might not meet your requirements. You then proceed by either failing to do so (all queries you can think of remain within your performance requirements) or succeeding to do so and choosing a course of action (optimize the proof of concept, or consider the concept a failure and go back to the drawing boards). If you’ve passed this step (either by optimizing your proof of concept or failing to produce queries that do not meet your requirements; or perhaps by tailoring your requirements  it happens) you verify that in all your proof of concept code, you have not used any MySQL code that would render your implementation Free Software. You have thus proven your third assertion: “Using ODBC allows us to use the MySQL database without rendering our own source code GPL and without paying for a license”.
While building your proof of concept, you are creating code. The code might not meet all of the requirements productionlevel code would meet, but it should be a very good starting point. Proof of concept code should therefore be developed using the same standards as production code and should be conserved in a working form  i.e. it should, during the development of the production code, serve as your first tests.
So, for a more concise answer to “How?”:

Analyse your requirements

Produce a set of assertions that are testable as hypotheses and of which proof will be sufficient evidence that the proposed solution is valid for the problem at hand

Find a solution that you think will meet the requirements

Conceive of tests to test each assertion (and make sure all stakeholders agree that the tests test the assertions adequately)

Develop and perform the tests, stopping as soon as one of your assertions fail.
You should test the assertion most likely to fail first: you should Fail Fast.
Who?
The answer to this really depends on how you manage your human resources: as a software analyst, I’d say my responsibility is to come up with solutions to your problems and that would normally include proving that the solution I propose is feasible and meets the requirements it sets out to meet  i.e. that it solves the problem. I also happen to be a programmer some of the time, so I don’t mind programming some of the time. On the other hand, there is something to be said for having a programmer  someone who will ideally be involved in developing the final solution  code the proof of concept under the analyst’s supervision: the programmer will know what to expect when the time comes to implement the final solution, and will have a far better understanding of what requirements the solution should meet if he implemented the proof of concept himself. That way, the proof of concept takes a bit longer to develop, but the production code takes less time to develop.
So there’s a tradeoff, but from a business perspective, timetomarket will usually win, as it should. Which allows for a shorter time to market depends on the complexity and risk of the concept: higherrisk or more complex concepts usually require more involvement from the analyst.
Conclusions
Proofs of concepts are important: they save time, they save money and they allow you to build your products on a solid foundation, with a better understanding of both the problem and the solution. They require an investment in the (potential) solution early on, but that’s when investments have the highest return and when modifications are least costly.