Computers run software and store files. The software and files are both stored as huge strings of 1s and 0s. Binary math is discrete mathematics.
Networks are, at the base, discrete structures. The routers that run the internet are connected by long cables. People are connected to each other by social media ("following" on Twitter, "friending" on Facebook, etc.). The US highway system connects cities with roads.
Doing web searches in multiple languages at once, and returning a summary, uses linear algebra.
Google Maps uses discrete mathematics to determine fastest driving routes and times. There is a simpler version that works with small maps and technicalities involved in adapting to large maps.
Scheduling problems---like deciding which nurses should work which shifts, or which airline pilots should be flying which routes, or scheduling rooms for an event, or deciding timeslots for committee meetings, or which chemicals can be stored in which parts of a warehouse---are solved either using graph coloring or using combinatorial optimization, both parts of discrete mathematics. One example is scheduling games for a professional sports league.
An analog clock has gears inside, and the sizes/teeth needed for correct timekeeping are determined using discrete math.
Machine Job Scheduling: Scheduling tasks to be completed by a single machine uses graph theory. Scheduling tasks to be completed by a set of machines is a bin-packing problem, which is part of discrete optimization. Google describes the issue for multiple types of jobs on multiple machines.
Cell phone communications: Making efficient use of the broadcast spectrum for mobile phones uses linear algebra and information theory. Assigning frequencies so that there is no interference with nearby phones can use graph theory or can use discrete optimization.
Digital image processing uses discrete mathematics to merge images or apply filters.
Source:
http://www.mathily.org/dm-rw.html
