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A conductor spreads charge evenly along it's surface.
The E-field on a point on the surface of a conductor is perpendicular to that surface.
If a charged conductor touchs an uncharged conductor, the charge will spread themselves evenly among the two conductors -- the two conductors essentially form one large compound conductor
Both conductors will now have equal electric potential due to having distributed the charge evenly among the surface area of both conductors.
NOTE: You can justify this conceptually by thinking about a case where the E-potential is not the same. If the E-potential is not the same, then charge would flow from the higher potential to the lower potential until the E-potential across both conductors is the same.
A charged object brought near an uncharged conductor will cause the conductor to polarize, which separates the charges of the conductor into two. This makes the conductor attracted to the charged object.
A charged object brought near an uncharged insulator can cause the atoms inside of the insulator to polarize, which may make the insulator attracted to the charged object.