When the PC turns on , at first step , BIOS searches bootloader program and then run it.
A single core named BSP (Bootstrap Processor) will run bootloader.
Bootloader lead your system to run kernel of OS.
BSP is one of the CPU cores that is specified by hardware and physical layer. this core used for initialization and shutdown processes.
Actually , the BSP is responsible for initializing the system and for booting the operating system.
Other cores are activated only after the operating system is up and running.
Kernel of operating system (its main thread) run on the BSP core usually and it manages other cores to run threads of current process or run some processes together.
Also , Cores able to doing some atomic instructions. For each core , these instructions ensures that the shared memory bus (high level cache) has exclusive ownership already.
These instructions helps OS to manage all process and threads (according their priorities) by software mutex implementation.
For example :
Intel CPU supports a prefix instruction named "lock".
lock inc [ebx]
A core that is running this instruction gets the bus immediately and other cores work will be suspended (until it is doing).
I do not know any things about multi-CPU systems. But , in the multi-CPU systems , each CPU has exclusive RAM and all CPU's connected via QPI channels (for Intel) together. Probably , because of separated RAMs , concurrency issues is less.