Corresponding objects are those that appear in the same place in two similar situations. It often happens with angles as shown below. Angle A on the left is the corresponding angle to K on the right, because they are in the same location in the two similar shapes. We say A corresponds to K.

This terminology appears frequently in connection with congruent and similar shapes.
See for example:

With parallel lines

When a line cuts across two parallel lines it creates pairs of corresponding angles as shown below. The two 70° angles are called corresponding angles.

See Corresponding angles for more.