Initializing array of structures

There’s no “step-by-step” here. When initialization is performed with constant expressions, the process is essentially performed at compile time. Of course, if the array is declared as a local object, it is allocated locally and initialized at run-time, but that can be still thought of as a single-step process that cannot be meaningfully subdivided.

Designated initializers allow you to supply an initializer for a specific member of struct object (or a specific element of an array). All other members get zero-initialized. So, if my_data is declared as

typedef struct my_data {
  int a;
  const char *name;
  double x;
} my_data;

then your

my_data data[]={
    { .name = "Peter" },
    { .name = "James" },
    { .name = "John" },
    { .name = "Mike" }
};

is simply a more compact form of

my_data data[4]={
    { 0, "Peter", 0 },
    { 0, "James", 0 },
    { 0, "John", 0 },
    { 0, "Mike", 0 }
};

I hope you know what the latter does.