ESP8266 Boot Mode numbers
When you power up your esp8266 a few numbers are printed to tell you in which mode the chip is right now.
Note: If you can't read the text via UART, try changing your baud rate to 74880 and reboot the board.This is what you should see in the logs:
ets Jan  8 2013,rst cause:2, boot mode:(1,6)
Reset cause and boot mode
Two things to note here: the reset cause and the boot mode.
From a forum post I've got the following:
reset causes:
    0: 
    1: normal boot
    2: reset pin
    3: software reset
    4: watchdog reset (if your board freezes)
boot source:
    0: maybe SD card? 
    1: ram
    3: flash
Boot Mode
That's an easy one:
- if you see 
boot mode:(1,x), that means the esp8266 is waiting for a firmware flash - if you see 
boot mode:(3,x)instead, your board is running your firmware. 
The x can change, but that's not relevant to know the boot mode.
TL;DR: boot mode 1 means wait for flash, while 3 means run the firmware.
Reset Cause
Usually it shows 2, even if you unplug the power and plug again. But can also be something else, as shown in the table above.