A packet is the unit of data that is routed between an origin and a destination on the Internet or any other packet-switched network.
When any file (e-mail message, HTML file, Graphics Interchange Format file, Uniform Resource Locator request, etc.) is sent from one place to another on the Internet, the Transmission Control Protocol (TCP) layer of TCP/IP divides the file into "chunks" of an efficient size for routing. Each of these packets is separately numbered and includes the Internet address of the destination. The individual packets for a given file may travel different routes through the Internet. When they have all arrived, they re reassembled into the original file by the TCP layer at the receiving end.

A packet-switching scheme is an efficient way to handle transmissions on a connectionless network such as the Internet. An alternative scheme, circuit-switched, is used for networks allocated for voice connections.
In circuit-switching, lines in the network are shared among many users as with packet-switching, but each connection requires the dedication of a particular path for the duration of the connection.

"Packet" and "datagram" are similar in meaning. A protocol similar to TCP, the User Datagram Protocol (UDP) uses the term datagram.