When a phrase is interpreted by the judicial system to have a specific meaning it is called 'legal construction'. Some terms have a very clear construction. For example, there is little ambiguity as to what constitutes a tort. In many other cases terms are far less clear.
I was not able to find a clear articulation of what 'adhering' or 'aid and comfort' mean. In fact, an online legal dictionary claims that there is no clear construction of these terms. However, you can find a list of notable treason cases over at FindLaw which may provide some insight into related issues.
A Historical Perspective
According to an old article in the Yale Law Review (Warren, "What is Giving Aid and Comfort to the Enemy?". 1918.) the Constitution's treatment of treason is taken from 1315 Statute of Treason, an English law. That law had several categories of treason and the Founders selected only one category to be treason in America. It isn't definitive, but their choice may help us exclude some things that treason isn't.
You can find a copy of the Statute of Treason here.
So what isn't it?
The Founders implicitly chose not to include these as treason:
- Planning to kill a political leader
- 'Violating' the wife or virgin daughter of the executive
- Counterfeiting the national seal
- Counterfeiting coinage
- Actually killing a political leader
The actual Act specifically mentioned which leaders may not be assassinated. I took the liberty of generalizing a bit here.
The phrases 'adhering' and 'aid and comfort' do not have a specific legal meaning, despite being otherwise uncommon phrases in modern English. We can look to the Founder's choice of definitions for treason for a little guidance, but ultimately the definition is going to come down to legal interpretation in any given case. There just isn't much guidance for you.