I don't understand conditions :(

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Data I don't understand conditions :(

Hi there,
I have a very general question. I'm rather new to (bash) shell scripting and I don't understand how conditions work... I've read numerous tutorials but I don't get it. I really don't. Sometime what I do works, sometime it doesn't and that's frustating. So what's the actual difference between these three forms. I haven't so far been successful in using the last one - always get a "syntax error". As well, what are their operators ? When should you use "-eq" or "==" ?
if  [ ... ]               if [[ ... ]]               if (( ... ))

As an example, why does the first one works but not the second!?!
if [[ -n $a ]]          if [ -n $a ]

Also, how on Earth do you do stuff like (I know this doesn't work - but I've seen things alike) :
(( 2 > 1 )) && (( echo "2 > 1" ))

If you know where a nice tutorial exists on the matter, I'll be glad as well!
Many thanks

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