Equations that can be written in the form $p(x) = 0$ for a univariate polynomial $p$ of degree $4$ or $p(X_1, \ldots, X_r) = 0$ for a multivariate polynomial $p$ of total degree $4$. Questions that use this tag should usually also have the polynomial tag.
A (univariate) polynomial is quartic if it has degree $4$, and a (univariate) polynomial equation is said to be quartic if it can be put in the form $$p(x) = 0$$ for some quartic polynomial $p(x) := a x^4 + b x^3 + c x^2 + d x + e$; sometimes one further imposes the requirement $a \neq 0$. (Some authors instead use the term biquadratic, but others only use the term biquadratic polynomial for quartic polynomials of a certain special form.) A question should be marked with this tag if it involves equations of this type, including questions about solving quartic equations.
In 1540 Lodovico Ferrari showed that one can always solve for the roots of a quartic equation in radicals, but degree $4$ turns out to be the largest for which this is true: It follows from the Abel-Ruffini Theorem that one cannot solve in radicals for the roots of general polynomials of degree $\geq 5$, and investigations into this and related questions were central in the early development of Galois theory.
By the Fundamental Theorem of Algebra, any quartic polynomial has exactly four roots, counting multiplicity. The character of these roots is determined partly by the discriminant of the polynomial, a sextic polynomial in the coefficients:
\begin{aligned} \Delta = &256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e\\ &{}-27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de\\ &{}+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\ &{}-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2} \end{aligned}
One method for solving quartic equations uses the resolvent, a cubic polynomial naturally associated to the quartic.
Quartic polynomials arise naturally, for example, in determining the intersections of conic sections and Alhazen's Problem in optics.
A multivariate polynomial is quartic if it has total degree $4$, and a multivariate polynomial equation is said to be quartic if it can be put in the form $p(X_1, \ldots, X_r) = 0$ for some quartic polynomial $p$. For $r = 2$, the zero loci of such polynomials are called quartic plane curves; this term also applies to the zero locus in $\Bbb P^2$ of homogeneous quartic polynomials in three variables.
Varieties that arise as the solution set of such polynomials include the Klein quartic.
Abramowitz, M. and Stegun, I. A. (Eds.). "Solutions of Quartic Equations." $\S$3.8.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 17-18, 1972.
Carpenter, W. (1966). On the solution of the real quartic. Mathematics Magazine 39: 28–30. doi:10.2307/2688990.
Cardano, Gerolamo (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3
Quartic equation (Mathworld)
Quartic function (Wikipedia)
