To the very best of my knowledge there is no formal definition of "Founder's stock". It is at best a shorthand which people use, and it has no precise, well-defined meaning.
I'm guessing that your consultant means a certain ownership percentage at the pre-money valuation in case you guys succeed in raising funds. So if there is an investment later on, then your consultant would have obtained shares at the same time as the founders, and would get diluted equally pro-rata with the founders.
Paying a consultant in stock at the extreme early stage is ... very unusual. I'm not saying it is always wrong; but consider this very carefully. First off, it can potentially be lots of money you give away. Second, he becomes a part owner, and all the legal implications need to be considered -- what are his voting rights, what are his rights in capital emissions (as when you take on new investors), etc. And lastly, getting a qualified lawyer to work through this with you is expensive.
Note: This is not qualified legal advice, I am not a lawyer, and you should see a qualified lawyer before proceeding.
Simply put, it's the shares that are associated with the original issuance of stock shares to investors. The Founders are typically those people who are directly involved in the creation of the company, developing the IP (i.e. - software, processes, etc.).
There are generally two classes (types) of stock - Preferred and Common. Most investors (especially early stage) want non-dilutable stock (meaning that their percentage of ownership will NOT decrease as new investors come on board in later funding stages). These classes can have many different kinds of terms and conditions placed on them, and it's best to retain a knowledgeable attorney to help you think through the various scenarios & their implications.
Usually, founders and original investors do not want to "get diluted"* as new investors come on board, and they typically want to retain control and voting rights as well. Very often, this will need to change, as later stage investors typically want similar rights.
He is probably asking you for equity in your company that he hopes will not get diluted and will grown in value a the company grows.
*i.e. their shares are the numerator in the fraction where the total number of outstanding shares is the denominator, and they want to protect against an increasing denominator.
Founder stock usually means common shares, as opposed to stock-options. I believe this has nothing to do with common vs. preferred shares, which show up later, once an investor has put some money in the company. Preferred shares have special rights.
When you offer stock to an early employee or a consultant, giving real stock instead of options has some value: with shares, you are a real shareholder, with all the rights associated with it. With options, you don't actually own real shares until you exercise those options, which could take a while.