What happens to the temperature when a substance is melting?

For most substances, it's the same temperature. For example, ice melts at 0°C (273K), and water freezes also at 0°C.

A few substances, such as agar, have a hysteresis. For example, agar melts at 85°C; to solidify it again, you have to cool it down to about 32-40°C.