Conceptually, it's a relative term, suggesting more of a discount than fares that have less of a discount, but not precisely defined beyond that.

However, some airlines make distinctions about which fare classes get more or less credit in their frequent flyer programs (see previous post for AA example), can be upgraded, qualify for premium seating, etc., etc. In that case the distinction is usually based on the specific fare class, identified by a letter such as Y, H or M. A fare that is considered "deep discount" in this sense may not be significantly cheaper than one which is considered simply "discount." (In unusual situations it may not be less expensive at all.) There have been many threads discussing whether is is worth booking a higher fare class for $X more in order to get the benefits of moving out of the "deep discount" category, how to force a specific fare class in an online booking engine, etc. These discussions are of necessity specific to the rules and regulations of one airline's program, so they tend to be found in individual airline forums.