Short Answer
You probably want an Illinois or Nevada limited liability company (LLC) but there aren't enough facts in the question to determine the founders reasons for wanting an "association" or "society" or what kind of business model it would have, which is necessary to really know which option is best.
Long Answer
Options Available
The most common forms of associations of people conducting business for profit are:
partnerships, which is the default form if no other legal action is taken,
limited liability companies, which is the most common form of small for profit entity which can be either manager managed or member managed, and
corporations (usually making an election under Subchapter S of the tax code), including but not limited to "professional corporations" (a largely obsolete distinction).
There are other forms of entities such as limited partnerships, limited liability partnerships, limited liability limited partnerships, and limited partnership associations, all of which are taxed as partnerships and limited liability companies are, and co-operatives, which are taxed like C-corporations except that they get a deduction for co-operative dividends paid to co-operative members who have a non-investor-like relationship to the company (e.g. as customers or vendors to the co-operative).
But, all of these forms of business organizations are much less common.
Governance And Organizational Considerations
So far as I can tell from your question, the only difference between what you are contemplating and these entity types is that you like the sound of the names "association" and "society".
If your emphasis in desiring to be an association or society is that you don't want to conceptualize your business as one with transferrable ownership shares, a partnership or limited liability company, or perhaps a limited liability partnership, might fit your needs.
In these entities, the partners or members have ownership and control rights that are not readily transferrable, the governance rules are targeted at a small number of owners, and there is room to customize how profits and losses and distributions are shared rather than the pro-rata distribution per share method that is most common for a corporation organized on a for profit basis.
Liability Considerations
A partnership's partners have unlimited joint and several liability for the debts and obligations of the partnership, so usually, if liability is at all a concern, a choice other than a partnership is the best one.
Limited partnerships are partnerships in which some partners with management authority called general partners have unlimited liability, while other partners with no management authority have limited liability.
In limited liability entities like limited liability partnerships (LLPs), limited liability companies (LLCs), limited liability limited partnerships (LLLPs), limited partnership associations (LPAs), and corporations, only company assets are subject to the debts and obligations of the company unless an owner of the company has liability for some reason other than being an owner of the company (e.g. having personally engaged in wrongful conduct like negligently driving a company car).
LPAs were invented to fit a potential tax niche that turned out not be be needed, are very rarely used and have no real advantages over an LLC.
So, typically, a limited liability partnership, a limited liability company, or a corporation would be the entity form of choice and the differences between an LLP and an LLC are quite subtle.
Most people starting a new entity choosing between an LLP and an LLC chose an LLC.
The LLP form of entity is mostly used by entities that were originally partnerships (perhaps from a time when certain kinds of professionals weren't allowed to form corporations and there were no other choices available) and then converted to have limited liability protections with a minimum of disruption.
The LLLP form is usually chosen either as a conversion from a limited partnership, or in order to as a matter of governing statute, have non-voting limited partners which has some subtle advantages in big dollar estates for estate tax planning purposes.
Tax Considerations
All of this said, the most important issue is choosing a form of organization for a for profit business is usually tax treatment. The most desirable entity type from a tax perspective depends to a great extent on the nature of the business.
If the entity is to own significant assets that are likely to appreciate in value and are economically important to the firm's profitability, a limited liability partnership or limited liability company, which have the same tax treatment, would generally be preferred.
If the entity is an operating business whose assets tend to depreciate and which generates lots of current income, an S-corporation is usually preferred, because it can result in lower FICA taxation and because the tax accounting for an S-corporation is much easier than for an entity taxed as a partnership (including most limited liability companies), and also because 1099s don't have to be issued to corporations.
In some niche applications (for example, state legalized marijuana businesses) it can be desirable to have a corporation not making an election under Subchapter S of the tax code and instead taxed as a C- corporation. In the case of marijuana businesses, a lack of pass through taxation (which is present in partnerships, LLCs and S-corporations, except LLCs electing to be taxed as C-corporations) is desirable to insulate owners from the entity for purposes of tax liability, and because of how Internal Revenue Code § 280E works. It can also be a tax treatment that makes sense in some international business situations, and in fast growing businesses that reinvest all of their profits in the business since capital is scarce. But, usually, paying taxes on corporate profits at both the corporate level and again in the form of individual income taxes on dividends or indirectly through taxes on capital gains from the sale of share of the corporation at a profit, results in higher aggregate taxation than pass through taxation in a partnership, LLC, or S-corporation.
Place of Organization
As far as choice of jurisdiction goes, the biggest question is Delaware or not Delaware. Delaware limited liability companies afford the organizers greater freedom to set up the internal rules of governance for the entity than any other U.S. state, including the ability to waive rules that are mandatory in almost every other jurisdiction such as the fiduciary duties of the managers of the company to the company. But with this freedom comes a lack of much structure in terms of default rules when an unforeseen situation arises. Also, Delaware's filing fees are among the highest in the U.S. and if there is litigation over the internal affairs of the firm, this will often have to be done in Delaware courts with Delaware corporate lawyers, which is a desirable thing if you are a large, publicly held corporation, because Delaware has lots of case law and a specialized court for the purpose and money is no object, but a considerable burden if you are a small, closely held firm, because Delaware corporate lawyers are extremely expensive compared even to Chicago corporate lawyers, and it is often an inconvenient place to litigate disputes. Generally speaking, I would disfavor it for the kind of venture contemplated in the question.
There are pros and cons between organizing in Illinois and Nevada, but really, the most important issue is where it would be most desirable for the parties to litigate their internal affairs disputes if they arose.
Non-Profit Choice of Entity Compared
The most common kinds of non-profits are unincorporated non-profit associations, which are the default for non-profits in the same way that partnerships are the default for for profit entities. They are what you call a group of people acting as an organization for purposes other than profit if they take no action to receive official government recognition. Unions and political organizations are two of the most common types of unincorporated non-profit associations.
The other kind of non-profit is a non-profit corporation, which is a corporation organized for purposes other than profit. Most entities you hear about that are called "associations" or "societies" which are organized on a non-profit basis are non-profit corporations. Similarly, most municipal governments and public colleges and universities are organized as corporations, which are not for profit, although typically not organized under a generic non-profit corporation statute.
