Telepathy is an ability seen in EarthBound Beginnings, EarthBound and Mother 3. In EarthBound Beginnings, it is a PSI ability learned by Ninten and Ana that allows them to read the minds of people or animals. It is also needed to enter the world of Magicant with the use of the GGF's Diary and a pink shell, use the magical fountain at Magicant as an ATM, and solve the riddle one of the Magicant's Palace guards sets on Ninten.
In EarthBound, Ness and Paula automatically use this ability when talking to people. Ness automatically uses it when talking to animals to communicate with them and when talking to people such as Lier X. Agerate to reveal their thoughts. Paula shows a much greater ability, using it to contact Ness and Jeff when she is in trouble.
In Mother 3, Kumatora and Lucas can use telepathy. Kumatora uses it when first meeting Salsa, and realizes that he needs rescuing. Lucas uses it to communicate with Boney. It's worth noting that Lucas, or perhaps just the player, is able to read Boney's thoughts before Lucas learns PSI.