Significance of Supari
English Name : Betel Nut
Supari is considered as very pure property of living being. In any ritual, we need to call deities by chanting ritual verses, in such case it is called "Sthapana" means establishment. But if we do not have statue of the deity, Supari is used. Supari is the representation of the deity which we want to call but we do not have his statue. Supari is symbol in such cases for deity like Brahma, Yamadev, Varundev, Indradev. Sometimes in "Grahashanti" it also represents planets like Mangal, Rahu, Ketu, Surya, Guru etc.
In short, Supari is alternative for such absence. In some ritual supari also represents human being, for example, some ritual require wife to be present, but if she is away or dead, Supari is used.
faiths and blind faiths
People also give Supari to GOD, in such cases it is believed that Supari is hard so all hard feelings are surrendered to GOD, so GOD may convert it in soft ones and will bless them.
Significance of Pan
English Name : Betel Leaf
In Hinduism the concept of "Aasan" or seat is there. When some guest arrive we usually ask him to have a seat. So Pan is used as a seat to supari. You may also find statues of deities are also kept or put on pan in some Yajnas or rituals.
If you do not have betel leaf, you can also use "Asopalav" tree's leaf. In various regions tree may vary, I do not know, but the concept is to give seat to supari.
Betel leaf, if someone eats it, kills germs and is considered that Lord Shiva and Mother Parvati has fed them, so they are considered "Sanjeevani" or "medicine", so in ritual it is also a symbol of wish for healthy life.
Disclaimer : This answer is not copy pasted from anywhere, it is just based on my knowledge about rituals. One may get cited answer of this question, but I am not able to find any citation from Hindu scriptures