How do windshear warning systems work?
Modern commercial airliners have 2 different wind shear protection system:
1) predictive wind shear warning system
2) reactive wind shear warning system. 

These systems work independently. PWS (predictive wind shear) works with weather radar. Reactive wind shear works with FAC ( Flight augmentation computer) 

Basicly PWS can be switched ON and OFF from the WX-Radar  panel and it has some prerequisite for working : 
Ground speed should be above 30 KTS, WX-Radar should be ON, At least one engine working etc. (may differ with different types).

PWS scans 5 NM ahead from 0-2300 ft and 1500 ft below according to phase of the flight. It works like a doppler radar. It scans the movement of air mass in front of you and "predicts" something goes wrong and warns you according to how bad is the situation. Doppler frequency will shift significantly in case of a microburst so PWS will catch this shift and warn pilots with amber or red cautions. 

Reactive Wind shear system works with on board computers that takes real time data from all sources. This system will be active generally from 50 ft 1300 ft. (may differ) 

Reactive Wind Shear basiclly monitors aircrafts energy level. For Airbus; FAC is responsible for that. 
In computing the energy level FACs use different data from different sources: ADIRS - air, ground speed, vertical speed , slope. from other sources - wind gradient, vertical wind etc. 

It will be active when the a/c energy level goes below predetermined threshold. It will alter from amber to red if your energy losing trend is faster. 

You can read this document from airbus includes many general aviation information for more details : 
http://www.airbus.com/fileadmin/media_gallery/files/safety_library_items/AirbusSafetyLib_-FLT_OPS-ADV_WX-SEQ02.pdf

Hope it helps.