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I would like to understand what a measure is, can someone help me understand this concept?
Wikipedia defines a measure as "a segment of time defined by a given number of beats", but this doesn't really explain how we would get the measure of a song.
If I am given a song, how do I find its time signature (i.e its beats per measure)?
Let's take a simple music sheet: 
As you can see, the time signature is given, and it's 4/4. That means every measure has 4 beats of quarters. I assume you know about the duration of the notes.
Every measure/bar in the music sheet is defined by the vertical lines you can see in the image above. Between every two vertical lines, you must have notes whose duration add up to 4 quarters (1 whole, 2 halves, 4 quarters, 1 half and 2 quarters etc).
As per Wikipedia, the measures/bars are used in order to help us read music sheets:
Dividing music into bars provides regular reference points to pinpoint locations within a piece of music. It also makes written music easier to follow, since each bar of staff symbols can be read and played as a batch
In this question you can see how to find the time signature of a song by listening to it.
When you say, "if I'm given a song", I assume you do NOT mean that you have sheet music. Is someone humming? Are you hearing a song on the radio?
You can usually find the measures and the time signature by listening carefully for the 1-beats. The 1-beats are the beginnings of musical phrases within the structure of a song.
For example, if you're listening to "Happy Birthday to You", you can hear the emphasis at the beginnings of the measures on "happy BIRTH-day to YOU" which tells you that (a) the measures begin on BIRTH-day and YOU, and that the song has three beats per measure, which typically means a 3/4 time signature.
That's a very simplified explanation that should help you without a lot of music theory knowledge.
Besides the definition of a measure I would also mention the grouping of notes in each musical phrase within the confines of the time signature. For instance, say were in 7/8 time sig. The eighth note gets the beat and there's seven beats per measure. One can play with a triplet feel plus 1. So 123 123 1. 123 123 1. One can also play it as three quarter notes plus 1 eighth note. 12 12 12 1. 1 and 2 are tied so not separate hits per se. When you get into polymeters it gets very interesting bc you can have 5/8, 7/8 and 9/8 going all at once with every player playing in groups of 3 so it almost sounds synchronized except for the 1s of each measure occur at different times. Then there are polyrhythms which is very advanced. Most popular music forms use 4/4 exclusively. It's the easiest and most common time sig. That's why it's called common time. Identifying 3/4 is fairly easy. Its a waltz. 123 123 123 123. Songs in 3/4 include Metallica - Nothing Else Matters, Jimi Hendrix - Manic Depression and Weezer - Holiday, Foo Fighters - Home, Led Zep - Kashmir. A more advanced song would be Schism by Tool. It starts with two measures of 5/4 then one measure of 4/4. Then it goes into the verses which alternate a measure of 5/8 then a measure of 7/8 back to one measure of 5/8 then one measure of 7/8, etc. I played with a drummer for a few years who is a rhythmical genius. He has a degree in higher math and drums/percussion from Berklee. That's where we all met. He used to tap 3s with his left foot, 4s with his right foot, slap 5s with his left hand on his knee, 6s with his right hand slapping on knee then speak 7s! It has to be witnessed to be believed. Total limb independence Polyrhythms.
One thing to keep in mind is that classical notation of time signatures are relative and subjective. This means that if you don't have it in written form or if it hasn't been decided or agreed upon by yourself or the people you are playing with, then it can be interpreted in different ways, to some degree. This is because, in its basic form, time signatures are like fractions. For example, 4/4 time, can also be written as 2/2 time or even 1/1 (although not common). If you are not worried about writing it down then you should focus only on the number of beats. To keep it simple, most music is in 4 beats per measure. As mentioned before, try and determine where the accents of the music are and then count to 4 for each pulse or beat that you feel. If the music doesn't seem to fit in 4 then try counting in groups of 3, since that is the next common group of notes. This is like a waltz feel.
Then, to make things more complicated, you might have to consider the subdivision of each beat. Normally the subdivision of the beat is an even number like 2 or 4, but in some feels the beat is subdivided into 3, like a blues shuffle.
So, in general: 1. Find the beat or pulse of the music, 2. Determine the accented beat or pulse, 3. Determine whether the beats group together in 4 or 3,
In addition, 4. Determine if each beat feels subdivided into 2 or 3.
Do you have an example?
