Basically, GRUB bootloader is the software that loads the Linux kernel. (It has other uses as well). It is the first software that starts at a system boot.
When the computer starts, BIOS first run a Power-on self-test (POST) to check hardware like memory, disk drives and that it works properly.
Then BIOS checks the Master Boot Record (MBR), which is a 512 byte section located first on the Hard Drive. It looks for a bootloader (like GRUB). The hard drive's partition tables are also located here.
Then you'll be prompted by the GRUB menu which can contain a list of the operating systems installed (in the case of dual boot), or the different kernels installed in a Linux system.
When you choose which operating system or kernel you want to use, GRUB loads the selected kernel. The kernel starts init (
systemd), which is the first process to start in Linux. The init program then starts other processes like network services and other applications that you might have configured to start at boot time.
The boot process contains a lot of defined expressions and terms, way too much to add in a simple answer like this very basic explanation of the boot process and GRUB's role in it.
This YouTube video provides a pretty nice introduction to the linux boot process, including basic information on the GRUB Bootloader:
Understanding the Linux Boot Process - CompTIA Linux+, LPIC-1