At first I saw that a resistor was required to protect a led from burning. From what I understand, it presents enough resistance to ensure not too much current flows thru it, on towards the led. The resistor is always placed after the outgoing red wire, right before the led, so into the positive terminal of the led.
But lately I've seen a couple of tuts where they use the resistor on the negative terminal of the led and on the wire going to ground.
It makes absolutely no difference which side the resistor is on: it's a series circuit so the current is the same through both components regardless.
Current flows round the whole circuit simultaneously(*), it doesn't encounter one component
and then another, its all the same current.
(*) ignoring the speed of light, fast changing signals etc, ie on timescales over about
a microsecond.
I guess since I've always thought of current as a flow of electrons between terminals with different electric potential, I imagined electricity flowing from the positive terminal of the battery, thru the toy or flashlight and then back in the negative terminal.
You need to comprehend the concept of a circuit. The word is generally derived from "circle".
Electrons flow round the circuit. There is no start and no finish. If it is not a complete circuit, there is no flow.
Check out this stuff.
Yes but a circle doesn't imply direction of flow. It may very well be a circle, but if the flow is clockwise, the hand will always reach the 1 before it reaches the 2, :-).
Just to throw another curveball at you - electricity doesn't really flow from "positive" to "negative/ground" (conventional flow). That's actually a historical anomaly caused by one individual's (Benjamin Franklin) educated, but unfortunately incorrect guess (he had a 50/50 shot, I guess) that wasn't fully understood to be incorrect until the 20th Century, when "electron flow" was understood:
We've been living with this ever since. It honestly won't effect anything you, or I, or any other electronics hobbyist is doing - unless they are playing around with low level semi-conductor physics calcs or such. For everything else, you can basically ignore it (until you start thinking about certain circuits - even simple circuits - then you go "hmm").
...forget I brought it up.
Woah, this is huge! Does anyone else know about this! 
JK. I had actually read a bit about this in my solar energy experiences. Something about holes in the metal atomic matrix being what actually moved?
Anyway, thanks.