Question

if two objects at different temperatures are in contact with each other, what happens to their temperatures.

Answer 1

Okay, let me reword this.  Heat flows to areas that are cool, however, once the heated area begins to cool, the other area begins to warm up. Then there comes a point when there is no energy being transmitted into the two areas. When that happens the two areas obtain/are at the same temperature.

For example, you let the warm are escape your kitchen and go into the cold garage. You leave the kitchen door open and the heat continues to flow into the cold garage. After a while you will begin to notice that the kitchen feels noticeably cooler and the garage feels somewhat warmer. Then you wait a while longer and realize that both of the rooms feel like they are at the same temperature. They have then reached equilibrium.

Answer 2

-- When the objects come in contact, heat energy begins to flow from the
warmer one to the cooler one. 

-- The warmer one cools as it loses heat energy, and the cooler one warms
as it gains heat energy. 

-- The flow of heat energy stops when the temperatures of both objects are equal.