Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to improve the situation.

The definition of the quantum group I saw is that it is a Hopf algebra given by some explicit generators and relations. Though I have heard that at least the case of the quantum $sl_2$ was motivated by physics, this was not explicit enough. If someone defined the classical (i.e. non-quantum) Lie algebra $gl_n$ or the symmetric group $S_n$ using generators and relations rather than operators acting on a vector space or on a finite set respectively, such a definition would be equally unclear to me. The abstract approach to quantum groups as deformations of a universal enveloping algebras in the class of Hopf algebras is very useful, but still not very intuitive and explicit.

While any clarifying remarks would be appreciated, I can ask the following more specific questions. (1) In simple examples of quantum groups, such as quantum $sl_2, sl_n$, are there "natural" examples of their representations, like the standard representation of the classical $sl_n$, its dual representation and their tensor powers? (2) Are there mathematical problems which are not about quantum groups, but whose solution does require this notion?