Solving an optimization problem with an annealing machine means representing elements as an Ising model. The system leads to an answer where each element is either $+1$ or $-1$. This is very different from the method for continuous values, where any point on the curve of the energy function can be a solution. "Intermediate solutions" between a solution and another solutio do not exist.
For example, in our shift optimization example, we formulate the problem in the form of whether or not Mr. A will work on a certain day (say, Febrary 10th). This is a discrete problem since it can be represented by two values, $1$ or $0$. A bit more complicated, if we choose among three options, whether Mr. A will be in charge of the hall on Febrary 10th, in charge of cooking, or on vacation, this is also a discrete value. Such a task assigning something will be a discrete value, so this is the type of problem where the annealing machine can be very effective.
In some cases, however, there are problems you can choose to define as discrete or continuous for the convenience of the calculation. For example, cutting pizza is a continuous problem if you consider it as dividing the area of a pizza. However, in reality, we would like each person to divide the pizza at an angle of about 30 degrees, so if we include such considerations in the problem, we can say that it is a discrete problem.