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You gotta feel it!! And there's the issue of how many "beats" are in the measure, on paper, vs. how you feel, or count, or conduct it when you perform it.
A piece in 3/4 time will always have three quarter-notes' worth of music in each measure. But you may play some of those pieces in one beat per measure (example: Waltz of the Flowers from The Nutcracker). Or three beats per measure (example: AMazing Grace). Or six beats to the measure (example: 2nd movement of Beethoven's 4th symphony). And that's strictly down to how fast the piece moves.
Could you do the Tchaikovsky in 3? Sure, but it would either be 3 really fast beats (awkward to conduct and to follow), or it would be slower than most people would want it. ("Most"--there's no accounting for taste.) Could you do Amazing Grace in 1? Sure, but again, that's not how it's most comfortable to most people.
One of the trickiest pieces I've played lately was the Cello Concerto by Victor Herbert. It's in 3/4, but it's too fast to be entirely comfortable in 3, and not fast enough to be entirely comfortable in 1. One could argue that this was strictly down to our soloist's choice of tempo, but IMO that tension was a big part of what made the piece interesting.
The best way to explain is with some audio examples, but I don't have any at hand.
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