Networking
The program or device that starts a network exchange by sending a request, such as a web browser asking for a page.
Formal
The party in a network exchange that opens the connection and sends requests, following a shared protocol, to a server that waits and replies. The role is defined by who starts the exchange, not by the kind of machine.
In plain English
Like a customer walking up to a shop counter - the customer asks, the shop answers, and it is always the customer who starts.
In practice
In a municipality's citizen service, the program on a clerk's PC is a client - each time she looks up a citizen, it sends a request to the central case system and shows the reply.
Why it matters
Clients run on the devices people use every day and process whatever comes back to them, so a tricked or outdated client is often how an attacker first gets inside.