Strange that people keep mentioning 掌 these last few days.
Neither reading is listed in the official "common use kanji list". The only reading listed for 掌 is its onyomi ショウ, with no kunyomi listed for it at all. This isn't to say that all other readings are wrong... Only that other readings aren't common ones which every fully literate adult is expected to have learnt and know.
手の平 (てのひら) = flat(平 "ひら") of the hand(手 "て")
掌 (たなごころ) = heart(心 "こころ") of the hand(手 "た")
Both refer to the palm -- an English dictionary definition of palm being "flat of the hand" -- which is essentially the "heart" (in the meaning of "centre") of the hand.
I've seen たなごころ marked as an "obsolete" reading on some things I searched it on, and I'm fairly sure 手のひら is a more common term, but I don't think this question is wrong. There's also たなうら (うら as in 裏) apparently too:
http://dictionary.goo.ne.jp/jn/137825/meaning/m0u/January 20, 2016 at 5:28am