As we know, for X86 architecture: After we press the power button, machine starts to execute code at 0xFFFFFFF0, then it starts to execute code in BIOS in order to do hardware i
... At the end, OS code starts to run. For ARM architecture, what is the booting process after use press the power button?
This answer is mainly in the context or modern Cortex-A CPUs; there are a great variety of ARM platforms. However, for an ARM that is PC like (tablet, cell phone, etc) ...
The ARM CPU will fetch an instruction from either 0x0 or 0xffff0000 (for a Cortex-M, it is data as opposed to an instruction). Typical ARM SOC have some boot rom which uses this mechanism. For an end user, you need to consult a manual to determine how to get your code to run. Ie, there is a BIOS built in to many ARM SOC which use the vector, but you need to use something different to get your code to run.
Typically the ARM SOC will support multiple boot devices. The device is determined by some FUSE (set by a manufacturing tool) or by sampling pins. The pins will be CPU outputs in a running system, but have been pulled up/down to configure a boot device. Each boot device will have peculiar details; ROM is simple, but NAND flash, SPI flash, MMC, etc need some configuration details. These are also often provided by a on-chip FUSE and/or sampling pins. A small portion of the device maybe read to further configure the device.
For a deeply embedded ARM chip, it may only boot from on-board flash and this process is much simpler; but I believe from the context of the question you are referring to more advanced ARM CPUs. More advanced ARM systems have a boot loader. This is because the amount of code a ROM loader will load is often limited and/or restricted. It is also often complex to set up SDRAM and the boot loader may be structured to run from internal static RAM, which configures the SDRAM.
See: Why we need a bootloader
Running the OS has its own peculiar issues. For ARM Linux, it was ATAGS and is now devicetree. People may code there own boot loader or use one of the many open-source projects with u-boot being the most common. U-boots supports vxWorks, Linux, NetBSD, Plan9, OSE, QNX, Integrity, and OpenRTOS as well a binary images.
Many original ARM Linux devices supported a direct boot of Linux without a boot loader. However, Linux does not support this in the main line except for a few very old ARM SOCs/cores.