This is something I definitely plan on writing about because when we're talking to new clients, the question of "Well ... if I get the 5 workflow package, what can I do with that?" comes up. Some will assume that they can do the largest solutions known to man with those 5; others will assume that nothing can be done with just 5.
A lot of the questions I also get are around Enterprise vs Standard.
I don't think the question is ever "what is nintex" but the answer nearly always is "nintex"! When we discuss requirements in our business we don't have a technology in mind as that can sometimes lead to a "fudged" solution (if you know what I mean). Generally though, our users want the same thing, they want to be able to submit/upload something, have something happen to it, maybe some approvals, and then an output. Whenever they talk about it, I can immediately picture the workflow or form in my head. They say things like "if the manager rejects then it can be resubmitted" and I immediately think, "Ok, we are talking a state machine here"... if they say "only the team leaders can do x", I think SharePoint groups and maybe item level permissions. Nintex workflow enables business to translate their requirements into solutions. It's just amazing haha!
HA! Yes, some people will try to get the most out of the least and then some people will get everything and never use the majority of the functionality.This is where this question or topic stems from; what is it and what can we do with it? Nintex is really amazing and has a multitude of functionality. It is difficult to pinpoint one or two things that it "does" because it can do a lot! It is easy enough to be picked up by a non-technical user and allow them to create workflows and forms, but powerful enough to allow more developer-like users the ability to customize and build amazing things!
It really is amazing how we begin to approach problems and view the world we live in after being exposed to Nintex and the power if provides. I try to teach the business analysts exactly what you mentioned; think of the business requirements in terms of how we are going to build the workflow and form. I am a huge proponent of "the right tool for the job" and Nintex has many tools available for us to pick from! Once you understand what is possible, you begin to push the envelope and start asking "what else"!
Functional:
Nintex = Proces automating and Empowerment.
Technical:
Extending default (SharePoint) functionality without the need of a developer.
Cassy Freeman: Partly human, Partly Nintex 
Simple answer for me is "An efficient way of delivering Business Process Automation".
I illustrate development times using SharePoint Designer (A), .NET workflows(B) and Nintex(C) workflows and that usually sells the product quickly enough.
Formula for this being to calculate the difference in development costs for a (average) workflow and then see how many times that difference goes into the cost of a Nintex license. For me, based on experience it usually takes around 2 developed workflows with Nintex to have covered the cost of the 5 license pack. This depends on your estimations, and the cost of your developers.
However, the issue with some of the less forward thinking folk in the companies I've worked with is "Why do I need Business Process Automation?"
I'm usually approaching team saying, "Did you know you could automate that with Nintex?" Anytime I see someone in another group demonstrating something that could be streamlined with Nintex, I schedule a demo with them to show them examples of how I used Nintex to streamline something similar. Therefore, my answer about what is Nintex is usually around automation and streamlining of processes already in place.
For me Nintex is the opportunity to deliver a great variety of functionality to people and this fast but more than that it's empowerment.
I go to a new customer, see what they want to achieve and after a short briefing I tell them "Just let's start to build a first draft". They are mostly like "what? now? Don't you need to go program something or so?" and I can show them how to build it now and with a little training they can build it on their own. This empowers people to do something new, to be less dependant on others and to have fun building their own freaky solutions.
Besides of all the great functionalities, all the solutions and performance increases it is this what I love most: seeing people stepping into new spheres of making their business run without the need to always ask IT to buy or program something.
Might be a little pathetic but that's what I love.
It is always a great feeling when you can teach someone and then watch them grow with it!
Nintex provides users the ability to do just that, and as you said, do it without the need of an IT resource. I am not as close to the business as most of these other users, so it only makes sense to empower them with the know-how and tools to succeed.
This is one of the first steps that I do when developing solutions; start with the basics! I always approach an ask with the mindset of "replace/recreate the current version AS-IS" and then look for areas to make improvements.
All too often users get wild ideas on how they can improve their process, but many times, if they simply automate it, the areas that need improvement are quickly identified.
Speed to market! This is something that we say over and over again! With Nintex, we can quickly get something up and running in under a week (sometimes an afternoon...shh...don't tell people that)! While there may be a larger effort to replace the process or functionality with a different solution, Nintex and the features that it offers can easily outshine most solutions simply due to ease of use, cost over time, and speed to market!
Not only SharePoint functionality, but now moving more into any business process! It gets the job done without the need of a developer, but can be even more robust within the hands of one.
