What is true name of android (download/fastboot/odin) mode? i saw every where to write something and named it this mode as download mode or fastboot mode or odin mode ..? what is true name of this mode? in iphone we named DFU mode that like this mode in android and done. What is the true name?
It is Download mode you boot the device into to use Odin. I've included a bit more info if you wanted anything deeper.
 See this post &quot;What is Download mode&quot;

As for Download mode, for example ODIN is commonly used on Samsung devices to flash ROMs, those devices needs to be in download mode prior to flashing. However, ODIN is not the official way to do it, rather Kies, is the official route in upgrading the firmware, this is for Samsung devices for example.

You can boot into Recovery mode also to use your recovery to make backups, wipe data, flash ROMs directly on the device.
XDA Wiki on Android Recovery and here is a short extract.

Recovery is stored in a disk partition separate from the main Android partitions (boot/kernel, root/system). It contains its own Linux kernel, separate from the kernel of the main Android system. Because of separate kernel+recovery_apps, the device is able to boot into the recovery mode even if the main system is broken, allowing the user a set of tools to fix the device. The recovery is completely independent of the Android system.

 See here for the CyanogenMod wiki on Fastboot and an extract:

fastboot is a small tool that comes with the Android SDK (software developer kit) that can be used to re-flash partitions on your device. It is an alternative to the recovery mode for doing installations and updates.
Because fastboot mode can start on your device even before Android loads (and can even run when Android isn't installed at all), fastboot mode is useful for updating the firmware quickly, without having to use a recovery mode. In fact, it's frequently the preferred way to initially install the recovery image on many devices. Fastboot can also be used for developer operations like unlocking the bootloader of Google's Nexus devices.