The golden compass, also known as an alethiometer, is a gadget given to Lyra by the Master of Jordan College at Oxford. The device is powered by Dust and can be used to gain insight into any number of questions. As Farder Coram tells us in Chapter 7, "aletheia" is Greek for "truth," so "alethiometer" means "truth measure" (7.118). It's a bit ironic that the designated keeper of this particular alethiometer is Lyra, who is known for telling whoppers.
The compass' face is made up of three swinging hands and a bunch of symbols, all of which have different levels of meaning. Usually it takes a person years to learn how to read an alethiometer, even with the assistance of a guidebook, but Lyra can read the compass quite easily. She explains her relationship with the compass here:
It's almost like talking to someone, only you can't quite hear them, and you feel kind of stupid because they're cleverer than you, only they don't get cross or anything.... And they know such a lot, Farder Coram! As if they knew everything, almost! Mrs. Coulter was clever, she knew ever such a lot, but this is a different kind of knowing.... It's like understanding, I suppose.... (9.87)
We know that there are prophecies about Lyra, and her growing skill with this instrument of the divine proves that there is something special about her.