Long division calculates the multiple at each step; short division just finds the remainder. That doesn't seem that different but when you come to do it algebraically you need to do long division. I did both at primary school (age 10), but it wasn't until 6th form (age 16) that we used long division again, those who hadn't done it at primary struggled a little.

       083.28
         2.26...
       ___
    7 |583.00

That's short division (sometimes called the "bus stop method" in UK), top line is answer [quotient], next line is "remainders". You say "7 in to 5 won't go; 7 in to 58 is 56 [just from knowledge of times tables, it's 56 tens your dividing], with remainder 2 [write remainder down, usually as a superscript to the dividend]; 7 in to 23 goes 3, remainder 2 [write remainder down]". Now you have the answer 583/7 is 83 remainder 2; but you can continue and divide the 20 tenths by 7, and so on.

Long division:

       083.28
         2.26...
       ___
    7 |583.00 [<-dividend]
       56     [=8x7]
       --
        23    [2 from subtracting answer to 8x7 {8 is put in answer line}, 3 from the dividend]
        21    [=3x7]
        --
         2.0  [2 is remainder, 0 from dividend]
         1.4  [=2x7 is remainder, 0 from dividend]
         ---
          .60
          .56 [=8x7]
           --
            4 [is the remainder in 100ths]

What we're doing is taking five-hundred and saying can we divide that by seven-hundred, we can't. So then we say well how about fifty-eight tens, can we divide that by seven-tens. The tens cancel each other out ( 10/10==1 ), so 58/7 = 8r2 but this is really saying 580/70 is "80 lots of 7" and 20/7 left over. So now we add that 20 units to the 3 units we have already from the dividend we started with, so now we need to do 23/7. And so on ...

I'd do an algebra example but it's a pain in the arse just using ASCII.