This equilibrium constant section (Chapter 4 of the Physical Chemistry textbook) provides a brief look into the history of the discovery of equilibrium.  As noted in the text, the scientists Berthelot and St. Gilles first represented the equilibrium equation using concentrations while Guldberg and Waage used kinetics. Equilibrium is reached when the forward and reverse reaction are happening simultaneously and at the same rate.  A brief review of the two systems might clarify the role each has to play in the overall understanding of equilibrium as used today.

Kinetics focuses on
(1)   how fast a reaction is or its rate of reaction and
(2)   what is actually happening to produce the products, also known as the mechanism.
In consequence, the overall rate is determined by the slowest step even if that step is the formation of an intermediate that does not appear in the original equation and therefore has no known concentration.
While equilibrium is dependent on the rates of the forward and reverse reactions being equal, it does not depend on either the time or the actual mechanism of how it was achieved.  Therefore, kinetic equations like the ones used by Guldberg and Waage are not useful in discussing equilibrium.
Concentration on the other hand is dependent on
(1)   the number of species present and
(2)   the overall importance, or coefficient of individual species.
The modern understanding of equilibrium uses both of these. This relationship is described by two variables, the equilibrium expression and the equilibrium constant.
The equilibrium expression is an equation that relates the concentrations of key products divided by the concentrations of key reactants with each raised to the power of its coefficient.  The equilibrium constant, K, is a numeric value found by plugging in the appropriate values.  For example:

While future sections discuss how to determine which products and reactants appear in the equilibrium expression, the power of the equilibrium constant is worth further exploration.  The constant has two different forms, variable and numeric with each being useful in different ways.
The variable form of K is generic in the sense that it implies equilibrium of a specific reaction under specific conditions but does not imply anything about the equation itself.  Subscripts are used to give information about the type of reaction and/or the states of the species involved.
K         Generic, species can be entered in a variety forms.
K_c        All species must be entered as molar concentrations, generally all aqueous.
K_p        All species are gases and entered as partial pressures.
K_a        Reaction is an acid ionization with molar concentrations.
K_b          Reaction is a base dissociation with molar concentrations
K_sp       Reaction is for a solid dissolving with molar concentrations.
If the specific equilibrium system is known, one that has a subscript, reference tables can be consulted for specific numeric data.  The generic form though has limitless possibilities and there is no reference data available.  However, it is sometimes possible to manipulate the specific equilibrium systems and their values to determine the equilibrium constant of a generic equation.
While the numeric form is useful for many purposes, three are of particular interest.  First, the actual value of the constant shows which side of the equation is favored with K >1 favoring products and K < 1 favoring reactants.  The magnitude of the value also indicates how much it favors one or the other with values much greater or less than one actually going out of equilibrium.
Secondly, because it is a constant under specific conditions, concentrations can be changed or predicted using the equilibrium expression.
Finally, it can be related to free energy and therefore spontaneity using ∆G = -RTlnK.  Variations can also be used for systems that are not under equilibrium conditions as will be discussed in future sections.
Overall, this part of the textbook offers a wonderful review of the history of equilibrium and those that discovered it, with the addition of a look at the power of the equilibrium constant.