There are multiple problems:
- C strings cannot be compared with - ==as you do. You just compare the pointers, not the contents of the arrays. You must include- <string.h>and use:
 -   if (strcmp(s[i].name, s[j].name) == 0) {
      /* duplicate name */
  }
 
- Also note that - gets()must not be used as it may cause undefined behavior if the input is too long. As a matter of fact, attackers may take advantage of this flaw to execute arbitrary code. Use- fgets()or- scanf("%19s", s[i].name).
 
- Why do you define the - studentsarray with 10 entries and only use 3 in the rest of the- main()function?
 
Here is a modified version:
#include <stdio.h>
#include <string.h>
struct student {
    char name[20];
};
int main() {
    struct student s[10];
    int n, i, j, count;
    for (n = 0; n < 10; n++) {
        printf("Enter student name: ");
        if (scanf("%19s%*[^\n]", s[n].name) < 1)
            break;
    }
    count = 0;
    for (i = 0; i < n; i++) {
        for (j = i + 1; j < n; j++) {
            if (strcmp(s[i].name, s[j].name) == 0) {
                count++;
            }
        }
    }
    printf("\n%d students have same name\n\n", count);
    return 0;
}
EDIT: Your counting method is incorrect for more than 3 entries: if the 4 students have the same name, count will be incremented for pairs 0,1, 0,2, 0,3, 1,2, 1,3 and 2,3 hence 6 students with the same name (!).
Here is a corrected version that does not use strcmp() either:
#include <stdio.h>
struct student {
    char name[20];
};
int same_string(const char *s1, const char *s2) {
    while (*s1 == *s2) {
        if (*s1 == '\0')
            return 1;
        s1++;
        s2++;
    }
    return 0;
}
int main() {
    struct student s[10];
    int n, i, j, count;
    for (n = 0; n < 10; n++) {
        printf("Enter student name: ");
        if (scanf("%19s%*[^\n]", s[n].name) < 1)
            break;
    }
    count = 0;
    for (i = 0; i < n; i++) {
        for (j = 0; j < n; j++) {
            if (i != j && same_string(s[i].name, s[j].name)) {
                count++;
                break;
            }
        }
    }
    printf("\n%d students have same name\n\n", count);
    return 0;
}