The two numbers at the start of every piece are the time signature. The top number is the count: how many beats fit inside one bar. The bottom number names the note value that's getting called "one beat" — 4 means quarter notes, 8 means eighth notes, 2 means half notes. So 4/4 says "four quarter-note beats per bar," and 6/8 says "six eighth-note beats per bar."
That's the definition. But time signatures are also a feel. Bars don't just contain a count — they contain a pattern of strong and weak beats, and that pattern is what makes a waltz feel like a waltz and a march feel like a march. The exercises below give you the same simple snare pattern in five different meters so you can hear the shape change without anything else getting in the way.
One more thing: meters with an 8 on the bottom (6/8, 12/8) are compound. The eighth notes group in threes, and you usually feel a slower pulse — two pulses in 6/8, four in 12/8. We'll cover that in detail when we hit the shuffle and triplet feel; for now, just notice how different a bar of 6/8 feels from a bar of 3/4, even though both contain six eighth notes' worth of music.