Let's start with the chip and work backwards. I'm simplifying some of this so it may not all be exactly technically correct.
Chip level - Machine Language. Binary numbers that are most frequently expressed by their hexadecimal equivalent.
Each of these numbers is an 'instruction' that tells the chip to do some very simple operation.Low Level - Assembly Language. Short alphabetical sort-of English instructions that correspond to the machine language instructions. There is a near one for one relationship between the assembly language instructions and the machine language instructions. The programmer uses a text editor to create a list of these assembly language instructions along with some other information which is then saved as a .ASM file. An 'Assembler' program (or human) converts this list of assembly language instructions into a list of the corresponding hex versions of the machine language instructions and saves them as a .HEX file.
High Level - C++. More complex sort-of English functions that correspond roughly with the machine language instructions that the processor understands or with more complex tasks that require many machine language instructions. The programmer uses a text editor to create a list of these functions along with some other information which is then saved as a .CPP file. A 'Compiler' program converts this list of C++ functions into the corresponding hex version of the required machine language instructions. These are usually saved as a .HEX file just like the one created by the assembler.
Arduino - ???. One more magical step similar to that described for C++ which someone else will have to explain. The Arduino functions are expanded into C++ functions which are compiled by a more or less standard C compiler as described above. Remember the phrase "along with some other information" also mentioned above? The Arduino environment (for lack of a better term) takes care of most of this as well.
Don
Actually even the assembly gets compiled first, because not every instruction writen in assembly exists, for example you can just type mul ..... even for micro-controllers that dont have and wired multiplier, but the compiler will generate the right code so you can do things like multiplications.
When coding in C, once again the code isnt directly passed from C to binary/hex code, it is first parsed, then optimized(in the case of the avr-gcc and avr-g++ compilers), and only then the optimized code is translated into assembly and finally to hex.
Optimizing compilers are great because they can really reduce your code size and make it faster, but it can make the debugging of said code a very painful thing.
The only thing that Arduino gives to you is simplified functions to interact with your avr, digitalWrite(13,HIGH), is in fact PORTB |= (1<<PB5), and the second runs about 40 times faster than the previous, because there is a lot of things done when you call the digitalWrite function.