I am curious how or if I could go about developing the product in partnership, or as part of an existing software company... Or maybe just selling it outright. I am building the prototype / proof of concept on my own, but I am not a developer, nor do I have the money to hire a developer on my own. I'm just trying to figure out what my options are.
If you believe your app could make a compelling addition to an established company, you can document it, and present it to the enterprise's decision-makers. You may first want to consult a patent lawyer if you're concerned about protecting the rights to your IP. Your other option would be to develop a business model around your app, and pitch it to a bunch of investors who may fund its development. Hope this helps!
Especially in software, where speed is key, you need to check demand. I like the LeanStartup methodology, everything you put on the canvas are hypotheses to verify, so detect the main ones and go check them out; with numbers, partners, allies or investors will come out of the stones.
I recommend using Invision, you don't have to make your app work, just hint at how it would work, kind of like the OZ wizard!
This is all about risk, not about your software product. All the actions you take to reduce the risk for a partner or buyer are what will sell the product, not your idea itself. Develop your marketing strategy and validate it. Test your assumptions. As @Luis said, go through the lean startup methodology to validate your product before you start shopping it around.
Not only will this give you an accurate framework to establish value if you try to sell the idea, it will inform you about what kind of relationship you might want to have to the product in the future (inventor, employee, founder, etc.).
You're likely to find that most people have no interest in partnerships that are fair. They may only be interested in an arrangement exclusively for their benefit. What you have to balance is what you want to walk away with, a job, an opportunity to sell more ideas, cash, or a symbiotic relationship. Consider what stage your plan needs to be in before someone can simply replicate your idea without you.