Basically, they are the concepts of energy (kinetic and potential) , and the concept of potential barrier, and the principle of conservation of energy. Also must understand differential equations (the schrodinger equation is used at the introductory level) and also understand very well the concept of the quantum mechanical wave function that appears in schrodinger equation.
The potential barrier means that in order for something to cross the barrier and due to the conservation of energy, must have energy (kinetic+potential) greater or equal to the energy level of the potential barrier. But it turns out that this is only if you treat the problem classically.
However using quantum mechanics and the schrodinger equation one can show that a particle can cross (to be more accurate, and having in mind the probabilistic interpretation of the quantum mechanical wave function, that the probability to find a particle with lower energy than the energy level of the barrier, at the other side of the barrier, is not zero as we would expect classically but it is positive (however small but positive)) the barrier having energy lower than that of the potential barrier and without violating conservation of energy.