Quantum computers are programmed by (evolving programming languages representing) so-called quantum circuits. These are a sequence of quantum gates plus the information on which quantum bits (qubits) they act.
The only thing you really need to know about quantum gates is that they represent rotations (in a higher dimensional space, so-called Hilbert space). Hence they are reversible: Quantum computers are programmed with reversible logic.
What do quantum gates rotate? It is the hyperspheres on whose surface qubit states live. Each state of the computational basis (|00⟩, |01⟩, |10⟩, |11⟩ for a 2-qubit system in the usual Dirac notation) gets a complex number as a coefficient or as a so-called probability amplitude. The basis vectors are orthogonal and span the state's Hilbert space, the probability amplitudes can be seen as coordinates in it. This is the picture in which quantum gates effect rotations. You will find that physicists often use a different picture, the Bloch sphere, for single qubit systems, in which quantum gates also cause rotations (but sometimes by a larger angle or by one that is omitted in that picture altogether).
All conventional logic can be implemented by a quantum computer by first expressing it in reversible logic (which may require ancilla bits). The classical NOT gate corresponds to the X quantum gate, but unlike the classical case where the only 1-bit reversible gates are the identity and the NOT gate, a quantum computer has four corresponding gates (X, Y, Z according to rotations on the Bloch sphere, plus the identity). Further, you can have rotations that rotate only by a fraction of how far these gates rotate; a few particularly interesting ones have special names and abbreviations such as the Hadamard gate or H gate that creates the equal superposition of all states.
Unfortunately, early quantum software engineers will probably have to know a bit about the quantum computer hardware they will be using: Due to the arbitrarily and continuously choosable angle of a quantum gate's effective rotation, there is a sort-of analog element to quantum computers that necessarily creates errors (and physical quantum computers have even more error sources than just that). There's a way to deal with it, quantum error correction that discretizes errors and corrects the most likely discretizations of them to achieve (ideally) arbitrarily complex computations with bound errors. But optimization will likely mean that one quantum computer with one choice of quantum error correction will be more apt at the certain quantum gate or even algorithms than others, in ways that are a bit more subtle than just looking up a speed factor.