"Tasbeeh is for men, and clapping the hands is for women." Is this an authentic Hadeeth? What is the meaning of this Hadeeth?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His Slave and Messenger.

The Hadeeth referred to is Saheeh (Authentic). It was narrated in the two most authentic books after the Quran: Al-Bukhari and Muslim. It was also narrated by Abu Daawood, An-Nasaa'i, At-Tirmithi, Ibn Maajah and others all.

The incident behind this Hadeeth was that the Prophet , sought to reconcile members of the tribe of Banu ‘Amr Ibn ‘Awf, and was late for the start of prayer. When he arrived, Abu Bakr was leading the prayer. When the Muslims who were praying saw the Prophet they clapped their hands to draw Abu Bakr’s attention. The Prophet beckoned to Abu Bakr to stay in his place. Abu Bakr   pulled back, so the Prophet came forward and led the prayer. After the prayer was finished, the Prophet taught the people the Hadeeth in question.

This Hadeeth teaches Muslims that when necessary the praying person can make a sound to alert the Imaam. If the prayer is being performed by a man, he may utter Tasbeeh, that is, to say, 'Subhaan Allaah', which means, 'Exalted is Allaah.' If the prayer is being performed by a woman, she may  clap by striking the palms of her hands together, or strike her left palm on the back of her right hand. It might be necessary for the praying person to do this to communicate with someone who is knocking on the door, to caution a blind person, or to alert someone who is absent minded.

Allaah Knows best.