ok let me try....saying it in 'plain English' without any use of musical terms is really hard, if not impossible. you need to at least know something about musical terms to understand what i'm saying.
The time signatures of every song is written right at the beginning of the score. It's beside the musical clefs (e.g. treble clef, bass clef, alto clef, tenor clef, etc.), where you see two numbers written one on top of the other. Sometimes it comes after the key signature - those '#'s and 'b's (the actual symbol resembles 'b', cos it isn't possible to type it out on the computer) that you see. It's hard to explain without a score. Take any music score and look at it. There will be time signatures like 6/8 (the '/' is used to replace the top-bottom alignment over here on the computer), 3/4, 4/4, (there are many more) and sometimes you'll see stuff like just a 'C' with/without the slash down its middle. the C alone stands for 'common time', which is equivalent to 4/4, and the C with a stroke down its middle is 'cut common time', equivalent to 2/2.
now on to what each time signature means. the top number stands for the number of beats in each bar/measure (those vertical lines that you see dividing the score into 'parts'). So a '2' stands for 2 counts, 3 = 3 counts and so on.
the bottom number tells you what note value is each count. a 4 stands for crotchet beats, a 2 stands for minum beats, an 8 = quaver beats, a 16 = semiquaver beats. so if the time signature is 6/8, it means '6 quaver beats in each bar' (though we commonly make it into '2 dotted crotchet beats in each bar' because it gives us the swinging feeling we need for pieces with this time signature). a time signature of 2/4 will mean '2 crotchet beats in each bar'. 4/2 means '4 minum beats in each bar'. and so on.
take note that you can't replace eg. a 4/2 with 8/4 or 2/1 (2/1 doesn't even exist). time signatures are not mathematical fractions, changing it would give the piece a totally different feel, even if the number of counts in each bar stays the same. same goes for 2/4. you can't replace it with 1/2 although both will give you 2 crotchet counts in each bar (a minum is twice the value of a crotchet).
it's hard to determine time signatures without any music scores, unless you've really good ears and can pick up on the number of counts + note values in each bar by just listening. a trained musician will be able to do some of this, because we're used to associating certain characters or 'feels' with certain time signatures. a waltz for example, is mostly in 3/4. a cradle song usually comes with 6/8 (the swinging feeling). a march is usually in 2 (2/4 or 2/2).
sometimes, the composer of the song may decide to change time signature in the middle of a piece. to know the new time signature, you just need to look for other appearances of the top-bottom numerical figures in the piece (the time signature) to get it. time signature changes usually come with a change of character in the piece, a new section, or some composers like debussy and other contemporary composers may just use it to purposely 'blur' the pulse of the song. frequent changes of time signatures result in a more fluid feel to the whole piece, and you feel as if the piece is just flowing on and on very smoothly, without eg. the clear cut 1-2-3 1-2-3 with the first beat emphasised every time.
i hope i made some sense in all that...if you need more explanation you can always email me and i'll try and send you a scan of the first page of any music score to mark out all that i've said.