What happens when using a cellphone when charging? What happens when a phone is being used with the charger connected?

Will it take power from charger and save the battery usage?

OR 

At one end, the charger keeps charging the battery and at the other 
end, phone discharges the battery simultaneously?

(I am using Sony Xperia SP with Android 4.2.1)
As using the device also uses some power, the device cannot "charge with the normal speed" at the same time. It will however primarily use the external power source for its consumption. Depending on whether you "consume" more power than the source can feed, either it compensates by taking whatever is needed additionally from the battery, or feeds whatever is left to it.

A side effect described in the other answers already is true, however: the device will get "warmer" this way. I already described that in another answer:


  Recommendation for cold days to get warm fingers: Take your device with both hands, plug in the charger. Now stream a HD video from youtube via 3G, play it fullsize, and in the background do some video recording which in parallel gets uploaded via 3G. CAREFUL !!! Wear gloves... #-)