Taker:
When you place an order that trades immediately before going on the order book, you are a taker. This is regardless of whether you partially or fully fulfill an order.
Trades from market orders are always takers, as market orders never go on the order book. These trades are "taking" volume off of the order book, and therefore are taker trades.
Maker:
When you place an order that goes on the order book partially or fully (such as a limit order placed via the trading screen on BIB Exchange), any subsequent trades coming from that order will be maker trades.
These orders add volume to the order book, helping to make the market, and are therefore termed maker for any subsequent trades.
Note: It is possible for a limit good ’til canceled (GTC) order (accessible via the API) to have traded as both a taker and maker.