char *p = new char[10]; dynamically allocates a memory block for an array of size of 10 chars and returns the address of its first element which is then stored into p (making p to point to the beginning of this memory block).
In this case, the new keyword is followed by an array type specifier, you specify the type:
char **p = new char*[10]; - type in this case is char*, not (char*). Check operator new[]
You are probably confused because of C-style malloc syntax where the type of its return value is always void*, which can be cast to different type so that you can dereference it. That's the situation where you use (char*) syntax (C-style type cast): char *p = (char*) malloc(10);
Although note that in C this cast is redundant: Do I cast the result of malloc?
Note that memory that was allocated by using new[] should be freed by calling delete[] and memory allocated by malloc should be freed by free.