The API Name of a field is the technical name for a field that is used behind the scenes to Identify all of the fields you have in your different systems. An API Name can also be called Internal Value, Internal ID or Item Name in the case of Sugar customers. These names are all lowercase and will not have any spaces between them. What you will see for the field name in the application is referred to as the field's “Label”.
Here's an example. You have a field called “Lead Source Detail” and this is the “Label” of the field (what you see in the application). But the API Name is “lead_source_detail”.