Does endianness have an effect when copying bytes in memory?

Endianness is only relevant in two scenarios

1. When manually inspecting a byte-dump of a multibyte object, you need to know if the bytes are ordered in little endian or big endian order to be able to correctly interpret the bytes.
2. When the program is communicating multibyte values with the outside world, e.g. over a network connection or a file. Then both parties need to agree on the endianness used in the communication and, if needed, convert between the internal and external byte orders.

Answering the question title.

Assume 'int' to be of 4 bytes

union{
   unsigned int i;
   char a[4];
};

// elsewhere
i = 0x12345678;
cout << a[0];   // output depends on endianness. This is relevant during porting code
                // to different architectures

So, it is not about copying (alone)? It's about how you access?

It is also of significance while transferring raw bytes over a network!.

Here's the info on finding endianness programatically

no when working on the same machine you don't have to worry about endianess, only when transferring binary data between little and big endian machines

Basically, you have to worry about endianess only when you need to transfer binary data between architectures which differ in endianess.

However, when you transfer binary data between architectures, you will also have to worry about other things, like the size of integer types, the format of floating numbers and other nasty headaches.

Yes, you are correct thinking that you should be endianness-aware when storing or communicating binary values outside your current "scope". Generally you dont need to worry as long as everything is just inside your own program. If you copy memory, have in mind what you are copying. (You could get in trouble if you store long values and read ints).
Have a look at htonl(3) or books about network programming for some good explanations.

Memcpy just copies bytes and doesn't care about endianness. So if you want to copy one network stream to another use memcpy.