I am not a lawyer, and I’m certainly not your lawyer. I have worked with third-party publishers who have made heavy usage of the Open Game License, but I wasn’t in charge of license compliance. The following is my understanding of matters based on being an interested party who has been “around” these issues.
Your rights, regardless of license
Paizo has no legal control over, for example, the English language—they cannot stop you from using English words like “fighter” or “investigator.” They also have no legal rights to folklore and myths, so “pegasus” and “Asmodeus” aren’t theirs to control, since both of those come from ancient sources.
Paizo does have legal rights—mainly copyright, but also some trademarks—over their own original content. The text and images and design of their books, for examples, are all covered by copyright. “Pathfinder Roleplaying Game” is trademarked, as is the Pathfinder logo. Copyright also covers characters, settings, plots, and so on—so while you can use the name “Asmodeus,” if your character is too similar Paizo’s character by that name, you could get in trouble.
Importantly, under US copyright law, game rules cannot be copyrighted. The text used to explain those rules, however, can be. This is a complicated area that would require extreme care to navigate successfully.
Additionally, in a select few cases, you can legally use copyrighted material—this is what is known as Fair Use in the United States, and pretty much everywhere else has something similar. Fair Use will almost-certainly not apply to any product you make, even if it’s free—Fair Use covers stuff like educational purposes (including reviews and the like), parody, and a few other things, but it doesn’t cover making your own product.
But the really important thing here is that no one can tell you for sure what’s safe and what isn’t. Cease and desist letters—the usually first step for dealing with copyright infringement—are just letters, there are no rules on who can send one or when. Even if you are completely in the clear, Paizo could send you one—and if you ignored it, you would run the risk of being sued, and having to defend your position legally, and that’ll cost even if you win in the end.
The Open Game License
The Open Game License is an attempt to limit some of the uncertainty around using Paizo’s work—it says you can use some of their material, even if it’s copyrighted, so long as you agree to their rules. Those rules include things like “Product Identity,” things you agree to not use, which would include Golarion (where Pathfinder campaigns usually take place), Iomedae (a goddess), and so on.
The first rule of the Open Game License is that you have to read and understand the entire license. You cannot use it if you don’t understand it. It has specific requirements for using it (including a copy of the license itself in your book, including filling in “Section 15” with every source you use, and every source those sources use, and so on and so forth (it’s a pain). It also spells out what you may do with the license and what you cannot do with the license.
Using the Open Game License will allow you to use most mechanical details in Pathfinder—classes, monsters, and so on. You could reference not just the word “fighter,” but the actual stats and mechanics of the fighter class, the bravery or weapon training features or whatever. Likewise for the investigator or even things that are more specific to Pathfinder like “monk of the seven winds” or whatever. You would be able to use aatheriexa, which without the license might be dubious.
If in doubt, a decent approach to things is to look at Pathfinder SRD sites, like d20pfsrd.com—since those are based on the OGL, they can only include things in the OGL (be careful with the Archives of Nethys, though—as Paizo’s preferred partner for these things, they may allow that site to include things that go beyond the OGL). Each sourcebook will also spell out explicitly what in that book is Product Identity—stuff you can’t use—so that is going to be the final arbiter of what you can and cannot use. Note that you will have to do this for your own work, as well. One of the requirements of using the Open Game License is that you also license your own work as open game content.
It’s not impossible to use the Open Game License without a lawyer—it’s designed to avoid that requirement—but you do have to be really careful about getting it right. And the only way to really be sure that you have gotten everything right would be to talk to a lawyer. None of the teams I’ve worked with has had a lawyer on hand to deal with these issues, but that’s taking a risk. If Paizo had ever objected to our usage of anything and sent a cease and desist letter, we would have had to pull the book in question, which would be very expensive.