Constant Memory is used to store constant values.
The advantage of having separate Constant Memory is to reduce latency. It is used in only those situation, when multiple threads has to access same value. How constant memory reduces the latency, I will explain now. Suppose there are 32 threads in one block. Let all of them are using the same variable. Hence there will be 32 accesses from global memory. Now if we store the variable in constant memory. Then first thread will access the value of a variable and will broadcast this value to other threads in half warp. This value will be saved in a cache and will be provided to the threads of other half warp. Hence total accesses will be just one instead of 32.
Constant memory can be used for a variable by prefixing keyword __constant__ in the variable declaration(e.g. __constant__ int x).