i am no good at drawing so try this.
christmas tree lights are in series.
The supply goes in the left side of the lamp and out the right side, then in the left of the next lamp and out the right, you with me so far?
In the case of christmas lights a wire is run from the last light back to the begining so that you have 2 wires to connect to, but there is no reason why you could not have lights on this wire too.
As christmas tree lights are 12v (usually) by having 20 in series that maans that you have 20 (lamps) @ 12 v = 240v
so these can be plugged directly into the mains (mains being 240 v)
(i only said in on the left and out on the right because i t was a 50 / 50 choice it does not actualy matter which way is in or out)
Paralel is different. (your sockets are in paralell)
Imagine the same lamps.
Now imagine two paralel lines, you connect each lamp across each line (as opposed to in and out (series)
The advanges are er um sorry i cant think of any just now, but the reasons different things are done is to do with the "thing " you are using, you wouldn't want 240v on christmas tree lights would you.
Edited by breezer to add
I found this drawing