Braces ($var vs. ${var})
In most cases, $var and ${var} are the same:
var=foo
echo $var
# foo
echo ${var}
# foo
The braces are only needed to resolve ambiguity in expressions:
var=foo
echo $varbar
# Prints nothing because there is no variable 'varbar'
echo ${var}bar
# foobar
Quotes ($var vs. "$var" vs. "${var}")
When you add double quotes around a variable, you tell the shell to treat it as a single word, even if it contains whitespaces:
var="foo bar"
for i in "$var"; do # Expands to 'for i in "foo bar"; do...'
    echo $i         #   so only runs the loop once
done
# foo bar
Contrast that behavior with the following:
var="foo bar"
for i in $var; do # Expands to 'for i in foo bar; do...'
    echo $i       #   so runs the loop twice, once for each argument
done
# foo
# bar
As with $var vs. ${var}, the braces are only needed for disambiguation, for example:
var="foo bar"
for i in "$varbar"; do # Expands to 'for i in ""; do...' since there is no
    echo $i            #   variable named 'varbar', so loop runs once and
done                   #   prints nothing (actually "")
var="foo bar"
for i in "${var}bar"; do # Expands to 'for i in "foo barbar"; do...'
    echo $i              #   so runs the loop once
done
# foo barbar
Note that "${var}bar" in the second example above could also be written "${var}"bar, in which case you don't need the braces anymore, i.e. "$var"bar. However, if you have a lot of quotes in your string these alternative forms can get hard to read (and therefore hard to maintain). This page provides a good introduction to quoting in Bash.
Arrays ($var vs. $var[@] vs. ${var[@]})
Now for your array. According to the bash manual:
Referencing an array variable without a subscript is equivalent to referencing the array with a subscript of 0.
In other words, if you don't supply an index with [], you get the first element of the array:
foo=(a b c)
echo $foo
# a
Which is exactly the same as
foo=(a b c)
echo ${foo}
# a
To get all the elements of an array, you need to use @ as the index, e.g. ${foo[@]}. The braces are required with arrays because without them, the shell would expand the $foo part first, giving the first element of the array followed by a literal [@]:
foo=(a b c)
echo ${foo[@]}
# a b c
echo $foo[@]
# a[@]
This page is a good introduction to arrays in Bash.
Quotes revisited (${foo[@]} vs. "${foo[@]}")
You didn't ask about this but it's a subtle difference that's good to know about. If the elements in your array could contain whitespace, you need to use double quotes so that each element is treated as a separate "word:"
foo=("the first" "the second")
for i in "${foo[@]}"; do # Expands to 'for i in "the first" "the second"; do...'
    echo $i              #   so the loop runs twice
done
# the first
# the second
Contrast this with the behavior without double quotes:
foo=("the first" "the second")
for i in ${foo[@]}; do # Expands to 'for i in the first the second; do...'
    echo $i            #   so the loop runs four times!
done
# the
# first
# the
# second