#include<stdio.h>
#include<ctype.h>
#include<stdlib.h>
struct horse
{
    int age;
    int height;
    char name[20];
    char father[20];
    char mother[20];
    struct horse *next;
};
int main(void)
{
    struct horse *current;
    struct horse *first;
    struct horse *previous;
    char test = '\0';
    for(;;)
    {
        printf("请问想填写马匹的资料吗（Y or N）:");
        scanf(" %c",&test);
        if(tolower(test) =='n')
            break;
        current = (struct horse*)malloc(sizeof(struct horse*));
        if(current == NULL)
            continue;
        if(first == NULL)
            first = current;
        if(previous != NULL)
            previous->next = current;
        printf("请输入马匹的名字：");
        scanf("%s",current->name);
        printf("请输入%s的age：",current->name);
        scanf("%d",&current->age);
        printf("请输入%s的height：",current->name);
        scanf("%d",&current->height);
        printf("请输入%s的father名字：",current->name);
        scanf("%s",current->father);
        printf("请输入%s的mother的名字:",current->name);
        scanf("%s",current->mother);
        current->next = NULL;
        previous = current;
    }
    current = first;
    while(current != NULL)
    {
        printf("马匹的名字叫做%s，它的年龄是%d，它的身高是%d.\n",current->name,current->age,current->height);
        printf("他的爸爸叫做%s，他的妈妈叫做%s\n",current->father,current->mother);
        previous = current;
        current = current ->next;
        free(previous);
    }
    return 0;
}

struct horse *current;
current = (struct horse*)malloc(sizeof(struct horse*));
scanf("%s",current->name);
你的malloc出来的只有4个字节的空间，不是一个结构体空间，后面又对结构体内部成员操作，明显是非法的
要改成：
struct horse *current;
current = (struct horse*)malloc(sizeof(struct horse));

for(;;){test='\0';.......在for循环开头为test赋值。