The device in a computer which is driving the address bus and bus control signals at some point in time. In a simple architecture only the (single) CPU can be bus master but this means that all communications between ("slave") I/O devices must involve the CPU. More sophisticated architectures allow other capable devices (or multiple CPUs) to take turns at controling the bus. This allows, for example, a network controller card to access a disk controller directly while the CPU performs other tasks which do not require the bus, e.g. fetching code from its cache.
Direct Memory Access is a simple form of bus mastering where the I/O device is set up by the CPU to read from or write to one or more contiguous blocks of memory and then signal to the CPU when it has done so. Full bus mastering (or "First Party DMA", "bus mastering DMA") implies that the I/O device is capable of performing more complex sequences of operations without CPU intervention (e.g. servicing a complete NFS request). This will normally mean that the I/O device contains its own processor or microcontroller.