I haven't done assembly since college but i will try to explain as much as i remember it might be bit off.
INCLUDE Irvine32.inc
this line is used to import Irvine32.inc which is required for 32 bit programming.
.data
An assembly programs is made up of multiple sections (memory segments) the data section is used to allocate memory spaces and variables to be used in the subsequent sections
arrayVal DWORD 1,2,3,4,5,6,7,8,9,10
here we create an array which is 32bit signified by DWORD and initialized with values 1 to 10. With arrayVal pointing to 1.
counter = 0
We will come to this little later i have doubts about this :P
.code
main:
The code section and main, the above two lines roughly says we are done with declaring variables, all the following stated stuff is instructions you should execute as main function.
Before i explain the other following code you should understand that even though you created memory segments that hold data they can't be used for operations in assembly. you must use special locations called registers for them more info here. edx, eax and ecx are all registers used for special functions.
xor eax, eax
xor edx, edx
register basically store binary data, they can have data from old operations before so we reset them to zero by using on xor with themselves.(they are faster than setting them by zero explanation here).
mov ecx, LENGTHOF arrayVal
this command basically moves the length of arrayval to the ecx register which is typically used as counter.
So the general logic will be. Read one value store to ebx, add it to eax (our accumalator) and then get next arrayvalue into ebx and add it to ebx and so on and so forth until we have added all values.
L1: and loop L1
the loop is instruction is very special. It basically does two things jump to where we have mentioned in this case L1 and reduce ecx automatically until ecx is zero. and everything in between gets repeated until ecx and hence these lines
mov ebx, DWORD arrayVal [edx]
add eax, ebx
inc edx
gets executed for all values of array. but this begs the question why do you need counter in the first place in data section(maybe it is a reserved word for ecx i am not sure).
Call WriteDec
exit
now after adding all the values if you call WriteDec it prints it values from eax register to standard output and we are done . so we exit and end main.
There are some things that seems off and unnecessary but if you google around a bit you should understand more. This seems a good place to start. Maybe read a couple of books, since you seem to be a very fresh beginner.