I wrote the function. But the teacher told me that in the 3rd parameters of the std::count_if function it is necessary to pass the lambda to find out if the letter is vowel. 
I can not figure out how to transfer it there.
unsigned CalculateVowels(const std::string& str)
{
    const std::string& vowels = "aeiouAEIOU";
    unsigned count = std::count_if(str.begin(), str.end(), [](int index) {return str[index] == vowels[index]; })
    return count;
}
                Your lambda function is wrong.
It needs to check whether the current element from passed str matches any of the elements in vowels. You can use the standard algorithm std::any_of from <algorithm> header for this.
#include <algorithm> // std::any_of, std::count_if
auto CalculateVowels(const std::string& str)
{
    const std::string& vowels = "aeiouAEIOU";
    return std::count_if(
        str.cbegin(), // for each elements in the elements of passed `str`
        str.cend(), 
        [&vowels](const char element) 
        {
            // following checks `std::any_of` the `vowels` element
            // matches the element in the passed `str`
            return std::any_of(
                vowels.cbegin(), 
                vowels.cend(), 
                [element](const char vow) { return vow == element; }
            );
        }
    );
}
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If that too much for one line, break it into small pieces.
#include <algorithm> // std::find, std::count_if 
auto CalculateVowels(const std::string& str)
{
    const std::string& vowels = "aeiouAEIOU";
    // lambda to check whether passed `char` element is a match
    // of any of the `vowels`
    const auto isVowel = [&vowels](const char element_to_be_checked)
    {
        return std::any_of(
            vowels.cbegin(),
            vowels.cend(),
            [element_to_be_checked](const char vow)
            {
                return vow == element_to_be_checked;
            }
        );
    };
    // now simply `std::count_if` the element `isVowel`
    return std::count_if(str.cbegin(), str.cend(), isVowel);
}
Or like @DimChtz tried to explain in the comments, using std::find
or even better as @RemyLebeau suggested, using std::string::find
#include <string>    // std::string::find
#include <algorithm> // std::find, std::count_if 
auto CalculateVowels(const std::string& str)
{
    const std::string& vowels = "aeiouAEIOU";
    const auto isVowel = [&vowels](const char element_to_be_checked)
    {
        // return std::find(vowels.cbegin(), vowels.cend(), element_to_be_checked) != vowels.cend();
        // or using `std::string::find`
        return vowels.find(element_to_be_checked) != std::string::npos;
    };
    return std::count_if(str.cbegin(), str.cend(), isVowel);
}
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