Power Up Issues with MacBook Pro A1278
The smoke coming out of your Macbook would lead anyone to conclude that the problem system had either an:


Overheating battery (common on some model MBP's) 
Overheating CPU (not at all common unless there is a blockage or other external heat.)


As others have suggested, it is definitely time for you to triage and rescue whatever remains on the HDD and take it to the Apple Store for a further diagnosis. Here are some things you can do in the meantime


Verify if the battery is the problem. Take a careful look at the chassis of the aluminum unibody. Does it seem swollen in some areas? If you use one finger to press on an edge of the MBP, does it rock? Swollen batteries are a common indicator of battery malfunction.
If the battery is removable from the bottom of the MBP, go ahead and remove it. Attach the AC adapter and power on your MBP without the battery installed. This should mitigate your device from turning off because of the heat from a malfunctioning battery. Other things you can do is place your device on a cool, dry kitchen counter while running it without the battery. This will further allow you to use the computer without risk of running into the CPU thermal throttling timer (simple function: too hot -> turn off). 
Lastly, if the above tip worked, immediately move your HDD contents to a backup location. 


Please report back on your findings and mark answered if this helps.