How does a computer know whether an arriving frame contains an IP datagram or an ARP message?

The computer known whether an arriving frame contains an IP datagram or an ARP message by type field it is 800 for ARP and 806 + for IP datagram.

Explanation:

The Internet Protocol is used to sent data from one host or a computer to another over Internet. Each computer has at least one IP address that is used in unique identification of that host from all other hosts on the Internet. It is used by Ethernet.

Address Resolution Protocol is used to convert an Internet Protocol address to a MAC address in a local area network (LAN).