This Longest Common Subsequence (LCS) algorithm is a classic dynamic programming table-filling algorithm.

static size_t longest_common_subsequence_internal(const char *str1, const char *str2, matrix **mat)
{
    int i, j;
    const size_t alen = strlen(str1);
    const size_t blen = strlen(str2);
    *mat = matrix_create(alen, blen);
    if (*mat == NULL) {
        return 0;
    }
    for (i = alen - 1; i >= 0; i--) {
	    for (j = blen - 1; j >= 0; j--) {
            if (str1[i] == str2[j]) {
                matrix_set(*mat, i, j, 1 + matrix_get(*mat, i + 1, j + 1));
            }
            else {
                matrix_set(*mat, i, j, max(matrix_get(*mat, i + 1, j), matrix_get(*mat, i, j + 1)));
            }
	    }
    }
    return matrix_get(*mat, 0, 0);
}

size_t longest_common_subsequence_length(const char *str1, const char *str2)
{
    matrix *mat;
    size_t result = longest_common_subsequence_internal(str1, str2, &mat);
    matrix_delete(mat);
	return result;
}

char *longest_common_subsequence(const char *str1, const char *str2)
{
    matrix *mat;
    size_t lcs_len = longest_common_subsequence_internal(str1, str2, &mat);
    unsigned int i = 0;
    unsigned int j = 0;
    unsigned int k = 0;
    const unsigned int alen = strlen(str1);
    const unsigned int blen = strlen(str2);
    char *lcs = malloc(lcs_len + 1);
    while (i < alen && j < blen) {
        if (str1[i] == str2[j]) {
            lcs[k] = str1[i];
            i++;
            j++;
            k++;
        }
        else if (matrix_get(mat, i + 1, j) >= matrix_get(mat, i ,j + 1)) {
            i++;
        }
        else {
            j++;
        }
    }
    lcs[k] = '\0';
    matrix_delete(mat);
    return lcs;
}

Example program:

int main(void)
{
    char str1[] = "pAqBrCs";
    char str2[] = "wAxByCz";
    char *lcs;
    lcs = longest_common_subsequence(str1, str2);
    printf("The longest common subsequence is \"%s\"\n", lcs);
    free(lcs);
    return 0;
}

The longest common subsequence is "ABC"