Basically, there are 2 types of Winshear alerts. There is a Windshear Caution (sometimes called a Windshear Alert). This level is an indication of improving aircraft performance (entering the front edge of a microburst, for example). One would think "If the aircraft is doing better, then why have an alert?" Well, usually, what occurs next is why. A Windshear Warning is an indication of rapidly deteriorating aircraft performance.
To put it easily, if the airspeed starts increasing very rapidly, that is an Alert. If it starts descreasing rapidly, that is a Warning. In actuality, what triggers each level is based upon many factors, including airspeed, vertical speed, and height above the ground (see my attached picture).
In general operation, in a Windshear Alert, and amber "WINDSHEAR" is presented on my attitude indicator. The flight director will continue normal flight unless the pilot pushes the TOGA buttons to tell the flight control computers that they will go around, then windshear escape guidance is brought up.
A Windshear Warning puts a red WINDSHEAR on my attitude indicator, along with an audibel alert (see the other thread for that sound), along with automatic windshear escape guidance, along with stall protection pitch limits, along with the autopilot automatically disconnecting.
But, like any automatic system, the GPWS can be fooled into thinking there is windshear just by a gust of wind. False windshear alerts are seen occasionally. False windshear warnings usually are true, but not always. "According to the book" a windshear warning must always be followed, but a windshear alert can be disregarded at the agreement of both pilots that a microburst situation does not exist.