One thing to note is that there is a difference between regular arrays and associative arrays. In regular arrays (real arrays), the index has to be an integer. On the other had, associative arrays can use strings as an index. You can think of associative arrays as a map if you like. Now, also note, true arrays always start from zero. Thus in your example, you created an array in the following manner:
a = [];
a["1"] = {"string1":"string","string2":"string"};
a["2"] = {"string1":"string","string2":"string"}
Javascript was able to convert your string indexes into numbers, hence, your code above becomes:
a = [];
a[1] = {"blah"};
a[2] = {"blah"};
But remember what i said earlier: True arrays start from zero. Therefore, the javascript interpreter automatically assigned a[0] to the undefined. Try it out in either firebug or the chrome/safari console, and you will see something like this when you try to print "a". You should get something like "[undefined, Object, Object]. Hence the size 3 not 2 as you expected.
In your second example, i am pretty sure you are trying to simulate the use of an associated array, which essentially is adding properties to an object. Remember associated arrays enable you to use strings as a key. So in other terms, you are adding a property to the object. So in your example:
b["key1"] = {"string1":"string","string2":"string"};
this really means:
b.key1 = {"string1":"string","string2":"string"};
Initializing b =[] simply creates an array, but your assignment doesn't populate the array. It simply gives "b" extra properties. Hope this helps.. :-)