How can a lightweight client verify a payment without downloading the whole blockchain?

Question

Accepted Answer

Simplified Payment Verification (SPV) keeps only the block headers of the longest chain plus a Merkle branch linking your transaction to a block — proving the network accepted it, without validating every transaction.
The SPV client queries nodes until convinced it has the longest header chain, then obtains a Merkle branch connecting the transaction to the block it's timestamped in. It can't check the transaction itself, but a valid branch shows a network node accepted it, and blocks stacked on top confirm ongoing acceptance.
The trade-off: SPV is reliable while honest nodes control the network, but can be fooled by an attacker who overpowers it. Businesses taking frequent payments often still run full nodes for stronger, faster verification.
Go deeper:

Simplified Payment Verification — Bitcoin Wiki — how a client using only block headers plus a Merkle branch checks a payment, and where the trust trade-off lies.
 Bitcoin whitepaper, section 8 "Simplified Payment Verification" — Nakamoto — the original description of header-only verification.