Simplified Payment Verification is described in section 8 of the white paper. A light wallet downloads only block headers. It then checks that a transaction is included in a block using a Merkle proof.
That is cheap, but it trusts that the chain with the most work also follows the rules, and it leaks some privacy to the servers it asks. A full node checks everything itself.
Takeaway: SPV is a good start; your own node is the full answer.