Time signatures are a method of musical notation, intended to convey the rhythm of a piece of music. Most music consists of repeating musical phrases, which in turn are each made up of repeating patterns. By breaking down these patterns and phrases, its possible to describe the rhythm to someone who has never heard the original piece. Most people can count along with the rhythm of a piece of music, and once you know where to start and stop counting, you'll be able to work out the time signature to practically any song.
Settle into the rhythm of the music and identify the beat. You don't need to count off any numbers yet, but find the regular pulse of the song and mark it by tapping your foot or snapping your fingers. Continue until you can do this without even thinking about it.
Find the "one." This represents the start of each new bar of the music. It is usually marked quite clearly by the beginning of the melody, or the start of the lyrics. Alternately, if the lyrics or melody start a little later in the bar, the end of each phrase will usually be signified with a drum fill or a crash. Look for repetitions in the phrasing or melody, and try to pinpoint where each phrase begins.
Start counting on the "one," and keep up with the rhythm, counting each beat off as you were before. Keep going until the end of the phrase, then start again from one when the next phrase begins. For instance, the nursery rhyme "Frère Jacques" has four beats to each bar, with one beat for each syllable in the words "Frère Jacques." Try singing the melody but replacing the words with "One, two, three, four."
This technique will tell you the number of beats in each bar. Count the number of bars in each repetition, and place the number after the beats-per-bar, and you have the time signature. For instance, four repetitions of three-beats-per-bar would give you a 3:4 time signature, like in the Cat Stevens song "Morning Has Broken."