You'll need to expand your array to make more space. Right now, you only have room for 20 characters.
Try
int names[8][25];
That will give you space for 8 names, each name having up to 24 char's.
Then you can loop through your names input. It's fine if you have only fewer than 8 names. Your player_num variable, will control all the output and game play. Don't use i for the player_num, though.
Traditionally, i is a loop counter, and you want something much more descriptive for your other variable names. It just helps a lot.
Code:
#include <stdio.h>
int main() {
int i, player_num;
char names[8][25];
printf("\n\n");
//user inputs value of player_num, here, as you have now
for(i = 0; i < player_num; i++) {
printf("Enter the player's name: ");
scanf("%s", names[i]); //enters name and creates a newline <enter key>
getchar(); //removes the newline from the keyboard buffer
}
printf("\n\n");
for(i = 0; i < player_num; i++)
printf("\n%s", names[i]);
printf("\n\n\t\t\t press enter when ready");
getchar(); //holds the console window open until you press enter
return 0;
}
For very small amounts of memory, as a beginner, it's much easier to just have a fixed number of names made available, than to start working with dynamically allocating memory. That can best be done, a bit later in your education.