Simplified Payment Verification is a way for Bitcoin wallets to confirm transaction data in a lightweight manner. It can be considered similar in some ways to Segregated Witness, which means to segregate blockchain transactions to make the network respond more quickly. In the case of simplified payment verification, you are doing away with the need for a Bitcoin wallet to download and verify the entire blockchain anytime a transaction is conducted.
Prior to SPV a wallet would need to download and check the entire blockchain to verify that BTC is available to be spent. But with SPV, the wallet, sometimes referred to as a ‘lightweight wallet’ cross references the blockchain to verify that any Bitcoin you receive is actually available to be spent. This significantly lowers the computational power necessary for running a wallet, and allows mobile wallets to function, as well as reducing the strain a desktop wallet might otherwise place on your computer if it needed to download and check the entire blockchain for each transaction.