A quantum computer is defined as a machine that can implement an arbitrary "unitary evolution" (up to arbitrary precision). If you don't know enough math to understand what a unitary evolution is, think of it as a quantum computer doing basically any possible thing that you can do with a given number of qubits. As another comment said, the idea is to (ab)use this to do some operations on qubits such that they end up in a state where measuring it will give you an answer to something you're interested in. It has very little to do with classical programming.

The math on quantum computers checks out, it's "just" an engineering challenge at this point, and many are doubtful whether these challenges will ever be overcome to build a quantum computer of sufficient complexity.

Essentially, some "unitary evolutions" are complex to implement, as in requiring a lot of quantum "gates". This causes an accumulation of error and a whole lot of other problems, which limits the complexity of the calculations that can currently be performed.