Armory does not need to be running to receive bitcoins. Once you have a receive address, coins can be sent to it at any point in time. You really should learn how Bitcoin works before using Armory. These are very basic block chain concepts.
Armory's main feature is offline transaction signing. This allows you to create transactions with an online wallet (contains no private keys) and sign it with your offline wallet (air gapped PC with private keys). Again, these are things you should be very comfortable with before proceeding.
There are many places you can go to learn about Bitcoin and Armory. This forum itself can be a useful resource. Use the search function.
https://en.bitcoin.it/wiki/Main_Pagehttps://bitcoin.org/en/https://btcarmory.com/