How does an online money transfer actually work from a technical and process standpoint?

That is a pretty broad question. However, the basic premise of all online money transfers is based on Trust and Settlements. Trusts represent agreements between the disbursing parties and Settlement is the method in which they will reconcile their accounts (balances).

Some relevant reading for you...

• Faisal Khan's answer to How does the settlement of payments work in banks? Specifically, how do payment systems that are connected to multiple banks actually settle the amount between two banks?
• Faisal Khan's answer to What exactly happens when a wire transfer is made? How does the money travel from the sender's account to the recipient's account? How much time does each step take? What regulatory measures are in place, if any?
  
• Faisal Khan's answer to What is a Paypal account? Is it a kind of bank account which is maintained for demand deposits? __ 𝐏𝐥𝐞𝐚𝐬𝐞 𝐂𝐚𝐥𝐥 PayPal 𝐓𝐨𝐥𝐥 𝐅𝐫𝐞𝐞 𝐒𝐮𝐩𝐩𝐨𝐫𝐭 𝐍𝐮𝐦𝐛𝐞𝐫 ☏+❶►❽❺❻▁❹❼❺►❷❺❺❺ ☏?
  
• Faisal Khan's answer to How does a remittance money transfer work? What are all the steps and who is involved?
  
• Faisal Khan's answer to How does the payments ecosystem work? What does the payments workflow look like, from the merchant’s card payment to the acquirer? What are the roles of each of the main firms in the space, like First Data, PayPal, Neteller, Square, Visa, etc.?