In its simplest form, the concept of the "rule of law" means that we are subject to clearly defined laws and legal principles (rather than the personal whims of powerful people), and that those laws apply equally to all people, all the time. The idea that no one is above the law is a foundational principle of American jurisprudence, even if its implementation remains more aspirational than actual.
Of course, the precise meaning of the rule of law can vary depending on the context and the speaker. So here are a few textual definitions of the rule of law, and what they can mean in given contexts.
The Oxford English dictionary defines the rule of law as: "The authority and influence of law in society, especially when viewed as a constraint on individual and institutional behavior; (hence) the principle whereby all members of a society (including those in government) are considered equally subject to publicly disclosed legal codes and processes." While Black's Legal Dictionary shortens (and somehow complicates) that definition to "[t]he predominance that is absolute of an ordinary law over every citizen regardless of that citizens power."
Black's does expand on that notion, by highlighting "Four Pillars of the Rule of Law":
And the U.S. federal court system has a similar definition, stating: "Rule of law is a principle under which all persons, institutions, and entities are accountable to laws that are: Publicly promulgated, equally enforced, independently adjudicated, and consistent with international human rights principles."
While all these definitions may differ in exact wording, they are all designed to legitimize laws and legal principles. If people don't have faith that laws themselves are just, and that the laws applied to them don't apply to others (like the lawmakers), the entire legal system will appear invalid. But simply saying that all laws apply equally to all people all the time doesn't, by itself, make all laws just. As Anatole France famously said: "In its majestic equality, the law forbids rich and poor alike to sleep under bridges, beg in the streets and steal loaves of bread."
Of course, the phrase "rule of law" can be invoked in a variety of situations. A simple Google search today indicates that President Donald Trump's cabinet "undermines the rule of law"; that if Roe v. Wade is overturned, "we should worry about the rule of law"; and that "Republicans for the Rule of Law" have shown support for Representative Justin Amash, who called for Trump to be impeached. The idea that the rule of law forms a foundation for how we act as a nation is one that crosses idealogical lines.
In that way, the phrase "the rule of law" can often refer to how we react to a political situation, rather than any specific legal rule. Which is not to say the rule of law behind the rule of law shouldn't remain a guiding principle of American jurisprudence, just that we could probably do a better job of following it, no matter who is in power.