An Application-specific integrated circuit, is an integrated circuit (IC) customized for a particular use, rather than intended for general-purpose use. For example, a chip designed to run in a digital voice recorder or a high-efficiency Bitcoin miner is an ASIC.
To learn more about ASICs in general, see the Wikipedia article: en.wikipedia.org/wiki/Application-specific_integrated_circuit
In Bitcoin mining hardware, ASICs were the next step for development after CPUs, GPUs and FPGAs. Capable of easily outperforming the aforementioned platforms for Bitcoin mining in both speed and efficiency, all Bitcoin mining hardware that is practical in use will make use of one or more Bitcoin (SHA256d) ASICs.
Note that Bitcoin ASIC chips generally can only be used for Bitcoin mining. While there are rare exceptions - for example chips that mine both Bitcoin and Litecoin - this is often because the chip package effectively has two ASICs: one for Bitcoin and one for Litecoin.
The ASIC chip of choice determines, in large part, the cost and efficiency of a given miner, as ASIC development and manufacture are very expensive processes, and the ASIC chips themselves are often the components that require the most power on a Bitcoin miner.
While there are many Bitcoin mining hardware manufacturers, some of these should be seen as Systems integrators - using the ASIC chips manufactured by other parties, and combining them with other electronic components on a board to form the Bitcoin mining hardware.