Spiders referred to as ''True'' spiders are those that are members of the Suborder Araneomorphae. Tarantulas are members of the Suborder Mygalomorphae so they are not ''true'' spiders. The spiders of the family Pholcidae (cellar spiders/daddy longlegs) however are in the Suborder Araneomorphae so are ''true'' spiders. If you are thinking of harvestmen which are also commonly called daddy longlegs, they are not any kind of spider as they are in the order Opiliones ( although they are arachnids).
Edit- ''Members of the suborder Araneomorphae are distinguished by having fangs that oppose each other and cross in a pinching action, in contrast to the Mygalomorphae (tarantulas and their close kin), which have fangs that are nearly parallel in alignment.'' ''In Araneomorphae, the fangs slope towards each other, giving these spiders many more possibilities than Mygalomorphae, which can only bite top down.'' I cut that out of wikipedia since I thought they did a good job explaining.