Praise be to Allah.
Clapping during parties is one of the actions of jaahiliyyah. The least that can be said about it is that it is makrooh (disliked), but the evidence suggests rather that it is haraam, because the Muslims are not allowed to resemble the kuffaar. Allah says describing the kuffaar of Makkah (interpretation of the meaning):
“Their Salaah (prayer) at the House (of Allaah, i.e. the Ka‘bah at Makkah) was nothing but whistling and clapping of hands”
The Sunnah is, when the believer sees something that he likes or dislikes, to say ‘Subhaan Allaah (Glory be to Allaah)’ or ‘Allaahu akbar (Allaah is Most Great),’ as was narrated from the Prophet (peace and blessings of Allaah be upon him) in many ahaadeeth.
Clapping is prescribed specifically for women if something alarms them during the prayer, or they are praying with the men and the imaam makes a mistake in the prayer. In that case they should draw his attention to that by clapping, whereas men should do so by saying ‘Subhaan Allaah’, as was narrated in the saheeh Sunnah from the Prophet (peace and blessings of Allaah be upon him). From this it may be known that clapping on the part of men implies imitation of kaafirs and women, and all of that is forbidden. And Allaah is the source of strength.
Shaykh Ibn Baaz.
The Standing Committee was asked about men clapping when they play with children, or children clapping to encourage their classmates. They replied:
This clapping is not appropriate, and at the very least it is intensely makrooh because this is one of the characteristics of the jaahiliyyah, and because it is something that is done only by women, to draw attention to a mistake in the prayer. And Allaah is the Source of strength.
From Fataawa Islamiyyah, vol. 4, p. 332-333
Children can be encouraged by saying ‘Allaahu akbar’ if they do something that the watcher or listener likes, or one may use other suitable phrases, or raise one’s hands, or raise one’s voice in words of praise such as “Well done!” or “Excellent!” and so on. And Allaah is the Source of strength.