Packet switching is a method of breaking down and reassembling data. If you send an e-mail to your friend, your computer chops your message (the data) into small pieces (packets). Each packet has an address telling the devices that move it along the network where to send it.
Your computer sends the packet to a nearby router. That nearby router sends the packet to another router closer to your friend's computer. That router sends the packet to an even closer router. On and on it goes until the packet reaches your friend's computer.
The packets that make up your e-mail message may all take different paths to reach your friend's computer, but finally they all get there. Now your friend's computer reads the instructions contained in the packets and reassembles the data into its original state. Your friend can read your e-mail.