Lightweight client in computer is a piece of software or hardware that connect to a server. An internet browser is an example of a client because it connects to a website's server and requests it's content.
 But in the context of a Blockchain, it is a software that connects to other clients in a pair to pair manner thereby forming a network of sorts where each client is a node.
A light client needs to download the block headers of the Blockchain; reason being the block headers contain a piece of information called the Merkle Tree root.
  This Merkle Tree root resembles a finger print of all information on the Blockchain about account balances and smart contract storage. Therefore should any tiny bit of information change, the fingerprint changes as well.
  Currently, light clients have a trusted Blockchain checkpoint built into their code.