C++ string operators
Don't include unnecessary headers

We're not using &lt;vector&gt; or &lt;stdio.h&gt;, so drop them.  We only need &lt;ostream&gt; rather than &lt;iostream&gt; for the class, and we prefer &lt;cstring&gt; to &lt;string.h&gt; to place the standard functions in their proper namespace.

Avoid importing all of namespace std

It's a bad habit that will cause problems in bigger programs.  Just don't.

Add some compiler warnings

I get some useful diagnostics, worth heeding, with g++ -Weffc++:

165851.cpp:33:39: warning: ‘DynString&amp; DynString::operator+(DynString&amp;)’ should return by value [-Weffc++]
     DynString&amp; operator+ (DynString&amp; y) {
                                       ^
165851.cpp:54:34: warning: ‘DynString&amp; DynString::operator+(char*)’ should return by value [-Weffc++]
     DynString&amp; operator+ (char* x) {
                                  ^
165851.cpp: In constructor ‘DynString::DynString(const char*)’:
165851.cpp:12:5: warning: ‘DynString::c’ should be initialized in the member initialization list [-Weffc++]
     DynString(const char* input) {
     ^~~~~~~~~
165851.cpp: In copy constructor ‘DynString::DynString(const DynString&amp;)’:
165851.cpp:19:5: warning: ‘DynString::c’ should be initialized in the member initialization list [-Weffc++]
     DynString(const DynString&amp; dynstring) {
     ^~~~~~~~~
165851.cpp: In function ‘int main()’:
165851.cpp:95:23: warning: ISO C++ forbids converting a string constant to ‘char*’ [-Wpedantic]
     stringD = "The" + stringB + " Wide Web";
                       ^~~~~~~
165851.cpp:95:33: warning: ISO C++ forbids converting a string constant to ‘char*’ [-Wpedantic]
     stringD = "The" + stringB + " Wide Web";
                                 ^~~~~~~~~~~


The last two are easily fixed by changing operator+ to accept const char* instead of char*.

The usual way of fixing the warnings on operator+ is to implement operator+ in terms of operator+= (the latter returns a reference, but the former an object).

BUG - be careful using a null pointer as sentinel

What happens if you evaluate DynString() + DynString()?  In both cases, the internal pointer points to NULL, but you happily call strlen on both - that's Undefined Behaviour.  Here's some code that actually checks:

DynString(const DynString&amp; other)
    : c(nullptr)
{
    if (other.c) {
        c = new char[std::strlen(other.c) + 1];
        std::strcpy(c, other.c);
    }
}


But it's probably easier (safer) to always allocate memory:

DynString(const char* input)
    : c(new char[(input ? std::strlen(input) : 0) + 1];)
{
    if (input) {
        std::strcpy(c, input);
    } else {
        *c = '\0';
    }
}


And then all methods can safely assume c isn't null.

Add a move constructor

This one's easy:

DynString(DynString&amp;&amp; other)
    : DynString()
{
    std::swap(c, other.c);
}


Overload operator[]

Allow for both constant and mutable string objects:

char&amp; operator[](size_t i) {
    return c[i];
}

const char&amp; operator[](size_t i) const {
    return c[i];
}


I believe size_t is more appropriate than int as the type of the index.

Eliminate the memory leaks

The example program leaks:

==13517== Memcheck, a memory error detector
==13517== Copyright (C) 2002-2015, and GNU GPL'd, by Julian Seward et al.
==13517== Using Valgrind-3.12.0.SVN and LibVEX; rerun with -h for copyright info
==13517== Command: ./165851
==13517== 
Hello Worlt
Hello World
The World Wide Web
sh: 1: pause: not found
==13517== 
==13517== HEAP SUMMARY:
==13517==     in use at exit: 47 bytes in 3 blocks
==13517==   total heap usage: 12 allocs, 9 frees, 73,860 bytes allocated
==13517== 
==13517== 11 bytes in 1 blocks are definitely lost in loss record 1 of 3
==13517==    at 0x4C2C93F: operator new[](unsigned long) (in /usr/lib/valgrind/vgpreload_memcheck-amd64-linux.so)
==13517==    by 0x108FF3: DynString::operator+(DynString&amp;) (165851.cpp:38)
==13517==    by 0x108C62: main (165851.cpp:90)
==13517== 
==13517== 18 bytes in 1 blocks are definitely lost in loss record 2 of 3
==13517==    at 0x4C2C93F: operator new[](unsigned long) (in /usr/lib/valgrind/vgpreload_memcheck-amd64-linux.so)
==13517==    by 0x10915B: DynString::operator+(char*) (165851.cpp:59)
==13517==    by 0x108CD5: main (165851.cpp:95)
==13517== 
==13517== 18 bytes in 1 blocks are definitely lost in loss record 3 of 3
==13517==    at 0x4C2C93F: operator new[](unsigned long) (in /usr/lib/valgrind/vgpreload_memcheck-amd64-linux.so)
==13517==    by 0x1091EF: DynString::operator=(DynString&amp;) (165851.cpp:69)
==13517==    by 0x108CE7: main (165851.cpp:95)
==13517== 
==13517== LEAK SUMMARY:
==13517==    definitely lost: 47 bytes in 3 blocks
==13517==    indirectly lost: 0 bytes in 0 blocks
==13517==      possibly lost: 0 bytes in 0 blocks
==13517==    still reachable: 0 bytes in 0 blocks
==13517==         suppressed: 0 bytes in 0 blocks
==13517== 
==13517== For counts of detected and suppressed errors, rerun with: -v
==13517== ERROR SUMMARY: 3 errors from 3 contexts (suppressed: 0 from 0)


The main cause is:

    char *newArray = new char[strlen(y.c) + strlen(this-&gt;c) + 1];
    strcpy(newArray, this-&gt;c);
    strcat(newArray, y.c);
    delete[] c;
    c = new char[strlen(newArray)];
    this-&gt;c = newArray;


The error is more visible if you remove the redundant this-&gt; from the last line:

    delete[] c;
    c = new char[strlen(newArray)];
    c = newArray;


The first allocation is obviously leaked, and that line of code should be deleted.  The same error is present in several other methods.

Implementation of operator+:

I would write it like this, starting with the += that the other versions need:

public:
    DynString&amp; operator+=(const DynString&amp; other)
    {
        char *newArray = new char[strlen(c) + strlen(other.c) + 1];
        std::strcpy(newArray, c);
        std::strcat(newArray, other.c);
        delete[] c;
        c =  newArray;
        return *this;
    }

    DynString operator+(const DynString&amp; other) const
    {
        DynString newString(*this);
        return newString += other;
    }


We don't need operator+(const char*), as a C-style string will be promoted to DynString using the (non-explicit) constructor.  We do need an overload to take a C-style string as the first argument - but it doesn't need to be a friend, as it can be an ordinary non-member function instead:

DynString operator+(const char* x, const DynString&amp; other) {
    DynString newString(x);
    return newString += other;
}




Improved version

This code is tested and does not leak:

#include &lt;ostream&gt;
#include &lt;cstring&gt;

class DynString {
private:
    char *c;

public:
    ~DynString()
    {
        delete[] c;
    }

    DynString(const char* input = nullptr)
        : c(new char[(input ? std::strlen(input) : 0) + 1])
    {
        if (input) {
            std::strcpy(c, input);
        } else {
            *c = '\0';
        }
    }

    DynString(const DynString&amp; other)
        : c(new char[std::strlen(other.c) + 1])
    {
        std::strcpy(c, other.c);
    }

    DynString(DynString&amp;&amp; other)
        : DynString()
    {
        std::swap(c, other.c);
    }

    friend std::ostream&amp; operator&lt;&lt;(std::ostream&amp; lhs, const DynString&amp; a)
    {
        return lhs &lt;&lt; a.c;
    }


    char&amp; operator[](size_t i)
    {
        return c[i];
    }

    const char&amp; operator[](size_t i) const
    {
        return c[i];
    }


    DynString&amp; operator+=(const DynString&amp; other)
    {
        char *newArray = new char[strlen(c) + strlen(other.c) + 1];
        std::strcpy(newArray, c);
        std::strcat(newArray, other.c);
        delete[] c;
        c =  newArray;
        return *this;
    }

    DynString operator+(const DynString&amp; other) const
    {
        DynString newString(*this);
        return newString += other;
    }

   DynString&amp; operator=(DynString other) {
       std::swap(c, other.c);
       return *this;
   }
};


DynString operator+(const char* x, DynString&amp; other) {
    DynString newString(x);
    return newString += other;
}


#include &lt;iostream&gt;
int main()
{
    DynString stringA("Hello");
    DynString stringB(" Worlt");
    //UNO
    std::cout &lt;&lt; stringA &lt;&lt; stringB &lt;&lt; std::endl;

    //DUO
    stringB[5] = 'd';

    //TRIO
    DynString stringC = stringA + stringB;
    std::cout &lt;&lt; stringC &lt;&lt; std::endl;

    DynString stringD;
    //QUATRO
    stringD = "The" + stringB + " Wide Web";
    std::cout &lt;&lt; stringD &lt;&lt; std::endl;
}




Make the concatenation more efficient

Implementing operator+ by copying and appending is inefficient because it allocates memory for the copy that is immediately released during the append.  So we can revisit that method and eliminate the copy step.  I'd do it using an extra constructor:

private:
    DynString(const DynString&amp; a, const DynString&amp; b)
        : c(new char[std::strlen(a.c) + std::strlen(b.c) + 1])
    {
        std::strcpy(c, a.c);
        std::strcat(c, b.c);
    }

public:
    DynString operator+(const DynString&amp; other) const
    {
        return DynString(*this, other);
    }


We can now simplify operator+=, implementing that in terms of our new operator+:

void swap(DynString&amp; other)
{
    std::swap(length, other.length);
    std::swap(c, other.c);
}

DynString&amp; operator+=(const DynString&amp; other)
{
    Dynstring newString(*this, other);
    swap(newString);
    return *this;
}


Remember the string length

We can make every method more efficient, and also allow for embedded NUL characters in strings, if we store the length as a member:

private:
    std::size_t length;         // not including terminator
    char *c;

    DynString(const DynString&amp; a, const DynString&amp; b)
        : length(a.length + b.length),
          c(new char[length + 1])
    {
        std::memcpy(c, a.c, a.length);
        std::memcpy(c+a.length, b.c, b.length+1); // copies terminator
    }


Lean on the standard library for memory management

Instead of managing our own memory, we could make c be a unique pointer:

private:
    std::size_t length;         // not including terminator
    std::unique_ptr&lt;char[]&gt; c;

    DynString(const DynString&amp; a, const DynString&amp; b)
        : length(a.length + b.length),
          c(std::make_unique&lt;char[]&gt;(length + 1))
    {
        std::memcpy(c.get(), a.c.get(), a.length);
        std::memcpy(c.get()+a.length, b.c.get(), b.length+1); // copies terminator
    }


You'll see that we do need to sprinkle calls to get() where we need pointers to the contents



Improved, efficient, smart-pointer version:

(just the class; everything else unchanged from first "Improved version" above, except an additional include)

#include &lt;memory&gt;

class DynString {
private:
    std::size_t length;         // not including terminator
    std::unique_ptr&lt;char[]&gt; c;

    // Helper constructor - allocates but does not initialize
    DynString(size_t length)
        : length(length),
          c(std::make_unique&lt;char[]&gt;(length + 1))
    {}

    DynString(const DynString&amp; a, const DynString&amp; b)
        : DynString(a.length + b.length)
    {
        std::memcpy(c.get(), a.c.get(), a.length);
        std::memcpy(c.get()+a.length, b.c.get(), b.length+1); // copies terminator
    }

public:
    DynString(const char* input = nullptr)
        : DynString(input ? std::strlen(input) : 0)
    {
        if (input) {
            std::memcpy(c.get(), input, length + 1);
        } else {
            c.get()[0] = '\0';
        }
    }

    DynString(const DynString&amp; other)
        : DynString(other.length)
    {
        if (length != other.length)
            c = std::make_unique&lt;char[]&gt;(other.length+1);
        length = other.length;
        std::memcpy(c.get(), other.c.get(), length + 1);
    }

    DynString(DynString&amp;&amp; other) noexcept
        : DynString()
    {
        swap(other);
    }

    void swap(DynString&amp; other) noexcept
    {
        std::swap(length, other.length);
        std::swap(c, other.c);
    }

    friend std::ostream&amp; operator&lt;&lt;(std::ostream&amp; lhs, const DynString&amp; a)
    {
        return lhs &lt;&lt; a.c.get();
    }


    char&amp; operator[](size_t i) noexcept
    {
        return c.get()[i];
    }

    const char&amp; operator[](size_t i) const noexcept
    {
        return c.get()[i];
    }


    DynString&amp; operator+=(const DynString&amp; other)
    {
        DynString newString(*this, other);
        swap(newString);
        return *this;
    }

    DynString operator+(const DynString&amp; other) const
    {
        return DynString(*this, other);
    }


    DynString&amp; operator=(DynString other) noexcept
    {
        swap(other);
        return *this;
    }
};