In algebraic combinatorics, there is an important concept of a "$q$-analogue". Surprisingly often when you have a counting problem with a good integer answer, you realize that it can be refined to a (finite) generating function with an equally good polynomial answer. A simple example of this is the $q$-analogue of the number of permutations, which is of course $n!$. If you define the weight of a permutation to be $q^k$, where $k$ the inversion number of the permutation, then the total weight is then the important and beautiful formula

$$1(q+1)(q^2+q+1)\cdots(q^{n-1}+\ldots+1).$$

This expression is called a $q$-factorial the factors are called $q$-integers. Interesting q-analogues usually involve $q$-integers. Note that $q$-integers are closely related to cyclotomic polynomials: Every $q$-integer is a (unique) product of cyclotomic polynomials, and every cyclotomic polynomial is a (unique) ratio of products of $q$-integers.

Gaussian binomial coefficients are among the most important $q$-analogues.

The best way to think of a quantum group $U_q(\mathfrak{g})$ is that it is an algebraic $q$-analogue of a simple Lie group $G$ or its universal enveloping algebra $U(\mathfrak{g})$. For generic values of $q$, it has exactly the same (names of) representations as $G$ that tensor in the same way, plus (depending on conventions) possibly some other representations that are less important. But, what has changed is the positions of the sub-representations and the specific representation matrices. In general, when you see expressions such as integers, binomial coefficients, and factorials in the formulas for representation matrices, you see $q$-integers, Gaussian binomial coefficients, and $q$-factorials in the quantum group version. The only asterisk to this is the preferred convention of using centered Laurent polynomials (which may have half-integer exponents) rather than standard polynomials with non-negative integer exponents.

The only non-generic values of $q$ for quantum groups are roots of unity. In this case a new and also fundamental effect appears: The representation theory acquires features shared with representations of algebraic groups in positive characteristic.  They have been used to strengthen or at least clarify the representation theory of algebraic groups.

The main application of quantum groups: Topological invariants. Eventually when studying the representation theory of a Lie group, you consider tensor networks, i.e., invariant tensors combined with tensor products and contractions. Because of the extra non-commutativity of a quantum group (or any non-commutative, non-cocommutative Hopf algebra), an invariant tensor network of a quantum group needs to be embedded in $\mathbb{R}^3$ in order to be interpreted or evaluated as an algebraic expression. And then the remarkable outcome is that you obtain the Jones polynomial, when the quantum group is $U_q(\text{sl}(2))$, and its well-known generalizations for other quantum groups.

Quantum groups are the main algebraic way to understand the quantum polynomial invariants of knots and links; and quantum groups at roots of unity are the main algebraic way to understand the corresponding 3-manifold invariants.  In fact, this is closely connected to why they were first defined.