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I searched the bitcoin wiki and found references to coinbase, but no clear cut explanation of what exactly it is. I know that miners can tag their coinbases (that is how some sites tell who mined a block).

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Not to be confused with the Bitcoin startup Coinbase, which is a hosted (shared) EWallet + exchange. – Stephen Gornick Nov 26 '12 at 5:12
    
For questions about the company see the tag coinbase.com. – Murch Nov 7 '15 at 15:20

The term "coinbase" is used to mean many different things. But the two you're probably asking about are:

  1. The "coinbase transaction" is the transaction inside a block that pays the miner his block reward.

  2. Inside the coinbase transaction is a field that is called the "coinbase". It's the generation transaction's equivalent of a scriptsig. Since it doesn't claim any existing outputs, it needs no normal scriptsig. It's basically just a random value that the miner can use as an additional nonce. BIP 34 changes this a bit.

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There's also now a Bitcoin exchange/hosted wallet web site called coinbase.com. – gosmond Feb 21 '13 at 4:25

David has already given a good explanation of the term coinbase, but I'd like to give further details on the coinbase transaction.

The coinbase transaction is a special type of transaction.

  • Every block must have a coinbase transaction, other transactions are optional.
  • The coinbase transaction must be the first transaction of the block (it follows that there can only be one per block).
  • The coinbase's output is used to send the block reward, i.e. block subsidy plus transaction fees, to the miner's address.
  • The coinbase transaction can have an arbitrary input of 100 byte size. E.g. this was where the genesis block famously contained "The Times 03/Jan/2009 Chancellor on brink of second bailout for banks". This is also the field that may be used for the extranonce.
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