A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".
In mathematics, it is often (though not always) desirable to provide a "closed-form", an expression in terms of a finite number of known functions, usually free from the presence of operators such as infinite series, (in)definite integrals, recurrence relations, diagrams, etc. This often allows for quick computation of the objects under consideration and may provide more insight into their behavior.
As an easy example, for $n$ a positive integer the sum of the first $n$ positive integers is $1+2+\ldots + (n-1)+n=\sum_{k=1}^n k$; however, this can be expressed in a closed-form as $\frac{n(n+1)}{2}$, which provides an efficient way to compute particular cases and gives a hint at asymptotic behavior. In a similar vein, $1^2+2^2+3^2+\ldots+n^2 = \frac{n(n+1)(2n+1)}{6}$; Eduoard Lucas conjectured that this value is only a perfect square for $n=1,24$, an observation difficult to spot without an explicit formula to work with. Other cases include Infinite Series $\sum\limits_{n=1}^\infty\frac{(H_n)^2}{n^3}$, which expresses an infinite series as a combination of three values of an elementary function (the Riemann zeta function), and Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $.
That being said, sometimes other properties, such as recurrence, are more helpful for various purposes. For example, starting with $F_0=0$, $F_1=1$, a closed-form for the Fibonacci numbers is $F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$ . This gives us the asymptotic $F_n\approx \frac{1}{\sqrt{5}}\phi^n$, but to compute $F_{20}$ it is much easier to use the recursion $F_{n}=F_{n-1}+F_{n-2}$. The recursion is likely more useful if one is doing combinatorics as well.
