The difference between NFC mode on Android (e.g. card mode, P2P mode) On my Samsung Galaxy Note 4, there are two NFC modes: card mode and P2P mode.

What is the purpose of these modes? Is there any other modes in NFC?
There are 3 NFC modes.
Taken directly from official Android Developer - Near Field Communication,

Android-powered devices with NFC simultaneously support three main modes of operation:

Reader/writer mode, allowing the NFC device to read and/or write passive NFC tags and stickers.
P2P mode, allowing the NFC device to exchange data with other NFC peers; this operation mode is used by Android Beam.
Card emulation mode, allowing the NFC device itself to act as an NFC card. The emulated NFC card can then be accessed by an external NFC reader, such as an NFC point-of-sale terminal.


To make it more easier to understand, Android Authority has a down-to-earth explanation,

In order to determine what sort of information is to be exchanged between devices, the NFC standard currently has three distinct modes of operation for compliant devices. Perhaps the most common use in smartphones is the peer-to-peer mode, which allows two NFC-enabled devices to exchange various pieces of information between each other. In this mode both devices switch between active, when sending data, and passive states when receiving.
Read/write mode, on the other hand, is a one way data transmission, where the active device, possibly your smartphone, links up with another device in order to read information from it. This is the mode used when you interact with an NFC advert tag.
The final mode of operation is card emulation, whereby the NFC device can be used like a smart or contactless credit card in order to make payments or tap into public transport systems.