I would suggest from an ATC perspective problems in aircraft come in 3 types
1)Those that do not effect flight safety and the ability to manoever the aircraft normally and conform to all normal ATC instructions
2) Those that may at some future time effect flight safety and the ability to manoever the aircraft normally and conform to all normal ATC instructions
3) Those that have an immediate flight safety impact.
Examples from a light aircraft point of view aligned with above could be
1) EGT guage has failed
2) Alternator has failed. Comms exist now but will be lost in 30 mins
3) The engine is running rough
For para 1 issues there would not normally be any need to tell ATC. For para 2 items ATC should be informed after you have troubleshot the problem and decided on what you want to do. Obviously para 3 problems mean you immediately declare an emergency.
In any case simply tell ATC what aircraft system is malfunctioning. There is no reason to tell a big story. The most important part of your call is to clearly state what you require (eg I need to land on the nearest runway, I need to descend now) and state any restrictions (eg once established on final I cannot accept a go around). If you are debating whether or not you should declare an emergency, that in itself is reason to declare.