When you execute something synchronously, you wait for it to finish before moving on to another task. When you execute something asynchronously, you can move on to another task before it finishes.
In the context of operating systems, this corresponds to executing a process or task on a “thread.” A thread is a series of commands (a block of code) that exist as a unit of work. The operating system runs a given thread on a processor core. However, a processor core can only execute a single thread at once. It has no concept of running multiple threads simultaneously. The operating system can provide the illusion of running multiple threads at once by running each thread for a small slice of time (such as 1ms), and continuously switching between threads.
Now, if you introduce multiple processor cores into the mix, then threads CAN execute at the same time. The operating system can allocate time to one thread on the first processor core, then allocate the same block of time to another thread on a different processor core. All of this is about allowing the operating system to manage the completion of your task while you can go on in your code and do other things.
Asynchronous programming is a complicated topic because of the semantics of how things tie together when you can do them at the same time. There are numerous articles and books on the subject; have a look!