Let's see some examples:
Example #1:

This is the way music in
4/4 time is organized metrically. We quickly learn that beat
1 is the most important (probably because it helps us stay together
better in ensembles) and that beats 2, 3 , 4 are weak beats. Thus,
young students often perform these notes in this way:
You can hear beat 1 clearly
accented and that beats 2, 3, and 4 sound much softer. Now
let's hear it another way, using a different approach to "note
grouping."
The difference you are hearing
here is that beat 1 sounds like the end of a group of notes and beat
2 "starts" the next group. See how the notes are
grouped differently below.
Example #2:

Now in slow tempos it may
be appropriate to subdivide the groupings further; i.e., beat 2 "moves"
to beat 3, beat 4 moves to beat 1 etc. Here is how that would
sound:
And here is how it looks
with brackets grouping beats 2 & 3 and beats 4 & 1:
Example #3:

Reducing the notation to
eighth notes follows the same procedure. See the example below. First,
listen to the notes performed according to the beams:
Example #4:

Now if we apply the same
note grouping ideas expressed above, the example sounds like this:
And here is how the notes
look when bracketed with note grouping principles. In slow
tempos it may be appropriate to subdivide the groupings further as
suggested above.
Example #5:

But let's move on. If
we divide the beat further, into four parts, the principles remain
the same. First, look at the example of 16th notes below.
Example #6:

Here's how it would sound
if the beams govern the expression:
And here is how it sounds
using note grouping:
And again, see the notation
with the note grouping brackets for visual help.
Example #7:

If we move from duple to
triple meter, the principles still remain the same. Look at
the example in 3/8 time below:
Example #8:

If we accent beat 1 each
time and let beats 2 & 3 be weak, it sounds like this:
If we regroup it as above,
it sounds and looks like this:
Example #9:

Before moving on to the next
lesson, try performing each example both ways so the difference between
performances based on the "beams" and performances based
on "note grouping" is clear to you.
[You will find further information
on the use of note grouping in musical expression in: James Morgan
Thurmond's Note Grouping, Camp Hill, Pa.: JMT Publications,
1982.]