A time signature is simply the composer's way of telling you how s/
he is subdividing the measure.
So in 4/4 time, the composer imagines the measure divided into four beats, while in 8/8 time, the composer imagines the measure divided into eight beats.
The difference is less in the strict timing and more in the feel or pulse of the music.
Try counting these out loud:
4/4 time: "1-2-3-4-/1-2-3-4-"8/8 time: "12345678/12345678"2/2 time: "1---2---/1---2---" All three of these take up exactly the same amount of time, but to me, the 4/4 time feels steady, while the 8/8 time feels brisk and the 2/2 time feels stately.
In all three time signatures, a quarter note will have the same value—one quarter of the measure—but in 4/4 time, it represents the pulse of the music, while in 8/8, it's two pulses and in 2/2 time, it's half a pulse.
A difference in feel between 3/4 and 6/8 time can be even more obvious.
In 3/4 time, the measure is broken up into three distinct beats, as in a waltz.
In 6/8 time, by contrast, the measure is often broken up into two beats, each with a triplet feel.
Like this: 3/4 time: "ONE and TWO and TRE and/ONE and TWO and TRE and "6/8 time: "ONE two tre
FOR fiv
six/ONE two tre
FOR fiv six " If the composer writes three quarter notes in 3/4 time, it will feel like three notes on the beat.
But if the composer writes three quarter notes in 6/8 time, the second note may feel syncopated relative to the overall pulse of the music.
Update: Listen to the first twenty seconds or so of La Pistola y El Corazon by Los Lobos.
Is it in 3/4 or 6/8 time?
Depending on how you hear it, the music has a much different feel, with the lead guitars putting emphasis on different parts of the melody.
Listen a few times and try to hear it both ways!
music.stackexchange.com/users/36/Alex Basson
Adding to Alex's answer, There seem to be at least 2 factors in choosing a time signature
First way is disregarding
duplet/triplet feel one is how many beats there are in the bar, so often in the case of 7/16
it's all about just the number of 16ths in that bar.
in many prog rock passages and places where the time signature is constantly changing it's because the player wants to extend a repeated phrase by an odd amount
so they'll just stick the notes in the phrase and change the time signature bar to bar.
As a practical example, suppose you have a phrase of nine quavers, grouped 3 3 3.
One way often used by prog rockers is to alter one group to be different lengths in a passage(let's use the last group here), so perhaps it would go 333,336,333,332.
In time signature terms
what you would have there is 9/8,12/8,9/8,8/8, It doesn't make sense to define the 8/8 bar as 4/4 because it breaks the pattern.
The second way is considering the feel
This is where 3/4 and 6/8 commonly differ.
As you may know 3/4 is counted in duplets or 1+2+3+(3 groups of 2), whereas 6/8 is counted in triplets or 1+a2+a(2 groups of 3)
The same can be applied to 4/4 and 8/8, where the difference is that 4/4 is counted 1+2+3+4+(or 4 groups of 2) but 8/8 can be grouped as (3 3 2),(3 2 3) or (2 3 3)in
the example of 332 this would be counted 1+a 2+a
3
+ You could arguably use 4/4, but in this case 8/8 suggests the feel better. Also for extra fun, here is my favourite example on 8/8 time signatures
music.stackexchange.com/users/6629/Alexander Troup
There is one commonly-used time signature that is just wrong.
6/8 literally means that there are six beats in the measure, and an eighth note gets one beat.
Almost always there are two beats in the measure and
a dotted quarter note gets one beat.
But there is no readable way to write that.
2/(8/3)?
2/2.666?
music.stackexchange.com/users/113/Mark Lutton
In 4/4 time, the concept of pulse is assigned to the quarter note beat.
However, in 8/8 time, which would seem to be similar, the sense of pulse is completely open-ended.
Often time signatures in 8 have a dotted quarter note pulse, such as 6/8, 9/8, 12/8, but in a time signature like 8/8, which is not historically used, the composer is free to subdivide as they choose.
This could technically be any variation of 3+3+2.
Basically, the difference is in the subdivision.
music.stackexchange.com/users/85/cotroxell
As an arranger, I have once used 8/8 as a time signature.
I wanted the performance to emphasise the nature of the quaver as the pulse, while recognising that the melody was easier to recognise as 4/4 than 4/8, thus preserving its legato style.
8/8 also led to a saner tempo marking...
music.stackexchange.com/users/647/Iain Hallam
These days I pretty much play everything by ear, but was originally trained in strathspey and reel violin so realise I should know the answer to this, but I can't remember what, if any, the practical differences were.  Is there any difference between a score marked 4/4 and one marked 8/8? Or is that sort of thing only useful to indicate a time signature change within a piece of music?
Is there any real-world difference between time signatures such as 4/4 and 8/8?
music.stackexchange.com
<time-signatures>
music.stackexchange.com/questions/568
music.stackexchange.com/users/104/Doktor Mayhem
The difference is less in the strict timing and more in the feel or pulse of the music.  iin 4/4 time, the composer imagines the measure divided into four beats, while in 8/8 time, the composer imagines the measure divided into eight beats.   In 4/4 time, the composer imagines the measure divided into four beats, while in 8/8 time, the composer imagines the measure divided into eight beats.   The difference is less in the strict timing and more in the feel or pulse of the music.  
In 4/4 time, the composer imagines the measure divided into four beats, while in 8/8 time, the composer imagines the measure divided into eight beats.   The difference is less in the strict timing and more in the feel or pulse of the music.  