I'm planning a switch to a desktop wallet for my crypto accounts. My first choice is Exodus (exodus.io). I understand that because it's a desktop wallet, the machine it's running on obviously needs to be powered, and the application needs to be running. My concern is what happens to transfers to my wallet when it is offline? Is the transaction held in the cloud network, until my wallet becomes available online, or do I need to plan to run it on a machine that is always active?
The wallet application on your desktop does not actually play a direct role in being able to receive or hold funds. Transactions involving your public address will continue to occur independent of your wallet.
Your account balance lives on the Ethereum blockchain. It is stored as a value which is included with each block, and associated with your public address. Your wallet simply manages the private key(s) to access your public addresses and the funds associated with them. When you have those private keys, you are able to sign messages/transactions for that public address, which is essentially proof of ownership.
This same private key can be imported into any wallet or "wallet-like" service, and gain access to the same funds.
If you want to learn in more detail, you should read the following:
More narrowly, from a programmer’s perspective, the word "wallet" refers to the system used to store and manage a user’s keys.