Is ASIC like a dedicated server which you can buy and then use online to mine or it is actually hardware that is actually shipped to your address which then you can use to mine?
An ASIC is a chip -- a physical hunk of silicon with metal and wires attached. An ASIC miner is a device containing one or more ASICs that mine. Frequently, people write "ASIC" when they are referring to an ASIC miner.
When you buy a miner, it is generally sent to you and you connect it to your computer, but there are "hosting" services that will put your miner in their data center and you can access it remotely, and there are "cloud" services that install and run their miners and sell you shares of the revenue.