The operating system is the layer among the hardware and the programs you run. It provides programs a standard interface to the hardware; or else every program would need to include its own device drivers. It remains track of the file system and provides security measures to protect data.
The operating system also handles memory allocation, protects one application from overwriting the memory of other applications, etc. It provides each program an address space so programs don''t require knowing exactly where everything is in physical memory. The OS maps that address space to real memory or still to virtual memory on disk, as required.
It also handles process scheduling. While it looks like you can run many programs immediately, the operating system is actually just speedily switching between each process and giving everyone a little time slice. If you have multiple CPUs or cores, the OS manages which methods run on which CPU.
There are also many additional higher-level things that are included in "the operating system" for ease, but could also be separated devoid of much trouble. Things resembling the windowing system, network protocols, etc.