What is the difference between the three?

In specific, if I purchased a widget and then sold the widget to someone else, on my books would the original widget be an expense, a liability, or the cost of goods sold?

Cost of Goods Sold

When you first acquire the widget, it becomes an asset. If you paid cash, then you wouldn't record a liability. If you purchased the widget on account, then you would have entered that into accounts payable which is a liability.

1. Assuming you paid cash

Dr - inventory
Cr - Cash

2. Assuming you acquired the widget on account

Dr - Inventory
Cr - Accounts Payable

Assuming you are using a perpetual inventory system, when you sell the widget, you make 2 entries.

1.
Dr - Cash or Accounts Receivable
Cr - Sales Revenue

2.

Dr - Cost of Goods Sold
Cr - Inventory