What are Ethereum transaction receipts and what are they used for?

Question

I've seen mention in a few places that blocks store transaction receipts, but I haven't really seen an explanation of what they are and why they exist.

Answer 1

Relationship between Transaction Trie and Receipts Trie provides a good summary:

Transaction Receipts record the transaction outcome

Here is the Structure of a transaction receipt

blockHash: String, 32 Bytes - hash of the block where this transaction was in.
blockNumber: Number - block number where this transaction was in.
transactionHash: String, 32 Bytes - hash of the transaction.
transactionIndex: Number - integer of the transactions index position in the block.
from: String, 20 Bytes - address of the sender.
to: String, 20 Bytes - address of the receiver. null when its a contract creation transaction.
cumulativeGasUsed: Number - The total amount of gas used when this transaction was executed in the block.
gasUsed: Number - The amount of gas used by this specific transaction alone.
status: String - '0x0' indicates transaction failure , '0x1' indicates transaction succeeded.
contractAddress: String - 20 Bytes - The contract address created, if the transaction was a contract creation, otherwise null.
logs: Array - Array of log objects, which this transaction generated.

Take a look at the last two properties. A simple use of a receipt is to find out a new contract's contractAddress. A more advanced used for a receipt is with Proving the Existence of Logs to the Blockchain

The status indicates if the transaction succeeded or not (ie. it ran out of gas)