Steal buildings and plans from a planet. "Appropriate" is just bureaucrat speak for stealing something.
When you give your spies the assignment to Counter Espionage, what you're doing is telling them to foil plots far and wide. If possible, not only will the plot be foiled, but the opposing spies will be captured and imprisoned. It's also possible that they might be killed in the process, but spying is a dangerous game.
When you tell your spies to gather intelligence there is all kinds of information they can get for you, and what they get depends on where they are.
Hack Network 19
The spy assignment Hack Network 19 is used to cost a planet happiness, and if succesful results in an insulting message about the empire being posted on Network 19.
Insurrection is the thing you should fear most in the game. It is the ultimate ability of a spy, in that it will allow a spy to steal an entire colony from you. Don't worry, it isn't easy as long as you try to defend yourself you shouldn't have to worry too much. Here are the things you need to know. A spy cannot incite insurrection:
Spy Mission: Incite Mutiny
Incite Rebellion reduces the happiness on a planet. Usually one of the following will occur with an attempt at this mission:
If you tell your spies to sabotage infrastructure then they will set out to destroy buildings and ships on whatever planet they're on.
There are three main categories of assignments or missions for spies.