Q:

Does specifying array size for a user input string in C matter?

I am writing a code to take a user's input from the terminal as a string. I've read online that the correct way to instantiate a string in C is to use an array of characters. My question is if I instantiate an array of size [10], is that 10 indexes? 10 bits? 10 bytes? See the code below:
#include <stdio.h>

int main(int argc, char **argv){

    char str[10] = "Jessica";
    scanf("%s", &str);
    printf("%c\n", str[15]);

}

In this example "str" is initialized to size 10 and I am able to to print out str[15] assuming that when the user inputs a a string it goes up to that index. 
My questions are:

Does the size of the "str" array increase after taking a value from scanf?
At what amount of string characters will my original array have overflow?
.

A:

When you declare an array of char as you have done:
char str[10] = "Jessica";

then you are telling the compiler that the array will hold up to 10 values of the type char (generally - maybe even always - this is an 8-bit character). When you then try to access a 'member' of that array with an index that goes beyond the allocated size, you will get what is known as Undefined Behaviour, which means that absolutely anything may happen: your program may crash; you may get what looks like a 'sensible' value; you may find that your hard disk is entirely erased! The behaviour is undefined. So, make sure you stick within the limits you set in the declaration: for str[n] in your case, the behaviour is undefined if n < 0 or n > 9 (array indexes start at ZERO). Your code:
printf("%c\n", str[15]);

does just what I have described - it goes beyond the 'bounds' of your str array and, thus, will cause the described undefined behaviour (UB).
Also, your scanf("%s", &str); may also cause such UB, if the user enters a string of characters longer than 9 (one must be reserved for a terminating nul character)! You can prevent this by telling the scanf function to accept a maximum number of characters:
scanf("%9s", str);

where the integer given after the % is the maximum input length allowed (anything after this will be ignored). Also, as str is defined as an array, then you don't need the explicit "address of" operator (&) in scanf - it is already there, as an array reference decays to a pointer!
Hope this helps! Feel free to ask for further clarification and/or explanation.

