In BigchainDB, transactions are used to register, issue, create or transfer things (e.g. assets).
Transactions are the most basic kind of record stored by BigchainDB. There are two kinds: CREATE transactions and TRANSFER transactions.
A CREATE transaction can be used to register, issue, create or otherwise initiate the history of a single thing (or asset) in BigchainDB. For example, one might register an identity or a creative work. The things are often called "assets" but they might not be literal assets.
BigchainDB supports divisible assets as of BigchainDB Server v0.8.0. That means you can create/register an asset with an initial number of "shares." For example, A CREATE transaction could register a truckload of 50 oak trees. Each share of a divisible asset must be interchangeable with each other share; the shares must be fungible.
A CREATE transaction can have one or more outputs. Each output has an associated amount: the number of shares tied to that output. For example, if the asset consists of 50 oak trees, one output might have 35 oak trees for one set of owners, and the other output might have 15 oak trees for another set of owners.
Each output also has an associated condition: the condition that must be met (by a TRANSFER transaction) to transfer/spend the output. BigchainDB supports a variety of conditions, a subset of the Interledger Protocol (ILP) crypto-conditions. For details, see the documentation about Inputs and Outputs.
Each output also has a list of all the public keys associated with the conditions on that output. Loosely speaking, that list might be interpreted as the list of "owners." A more accurate word might be fulfillers, signers, controllers, or transfer-enablers. See BigchainDB Server issue #626.
A CREATE transaction must be signed by all the owners. (If you're looking for that signature, it's in the one "fulfillment" of the one input, albeit encoded.)
A TRANSFER transaction can transfer/spend one or more outputs on other transactions (CREATE transactions or other TRANSFER transactions). Those outputs must all be associated with the same asset; a TRANSFER transaction can only transfer shares of one asset at a time.
Each input on a TRANSFER transaction connects to one output on another transaction. Each input must satisfy the condition on the output it's trying to transfer/spend.
A TRANSFER transaction can have one or more outputs, just like a CREATE transaction (described above). The total number of shares coming in on the inputs must equal the total number of shares going out on the outputs.
Example 1: Suppose a red car is owned and controlled by Joe. Suppose the current transfer condition on the car says that any valid transfer must be signed by Joe. Joe could build a TRANSFER transaction containing an input with Joe's signature (to fulfill the current output condition) plus a new output condition saying that any valid transfer must be signed by Rae.
Example 2: Someone might construct a TRANSFER transaction that fulfills the output conditions on four previously-untransferred assets of the same asset type e.g. paperclips. The amounts might be 20, 10, 45 and 25, say, for a total of 100 paperclips. The TRANSFER transaction would also set up new transfer conditions. For example, maybe a set of 60 paperclips can only be transferred if Gertrude signs, and a separate set of 40 paperclips can only be transferred if both Jack and Kelly sign. Note how the sum of the incoming paperclips must equal the sum of the outgoing paperclips (100).
When a node is asked to check if a transaction is valid, it checks several things. We documented those things in a post on The BigchainDB Blog: "What is a Valid Transaction in BigchainDB?" (Note: That post was about BigchainDB Server v1.0.0.)