Hi Paul
Music is written in Bars, or 'measures'.
In a time signature such as 3/4 (which of course is slow waltz), the upper number tells you how many beats there are in the bar, and the lower number tells you the 'value' of each beat. For slow waltz it's a quarter note. Of course the definition of 'how long does a quarter note last' is often woolly. Classical composers used to specify this at the start of their score. It essentially defines the tempo of the music. If you play slow waltz at 30 Bars per minute, then each quarter note has a 'value' in time of 1/90 of a minute,(30 bars * 3 = 90 beats) or 2/3 of a second. Viennese waltz is also 3/4 time, but each quarter note is allocated less time. But there is no mistaking 3/4 time as 'waltzy'.
Now, of course, you can have more than 3 notes in a bar of waltz music. Imagine dancing a simple chassis (1,2&3 timing). The musical notation equivalent would be a quarter note (beat 1) , 2 eighth notes grouped together (2&) and another quarter note (beat 3). So 4 notes (4 steps) but 3 beats.
6/8 time would have 6 beats to the bar, each one with a time 'value' of an eighth note. A jig would be an example.
Remember, musical notation was written for musicians, not dancers, though there's no reason why dancers shouldn't be aware of it.
Accented notes: in all music, the first beat in the bar is usually accented(i.e. played slightly more strongly) because the first note of a bar would normally represent the start of a new 'theme' or 'idea' in the music.
In foxtrot, which is 4/4 time, beat 3 is also accented, but to a lesser extent. Thus, if you start a figure with a strong first step (as in the feather step ) it's normal to start this on beat 1 to match the stronger accent at the start of the bar. You can start it on beat 3, but it feels different - less 'musical'. Listen to Ella Fitzgerald singing 'night and day' or 'begin the beguine' and you will hear her playing with her timing as she sings. Similarly dancers play with the timing (within limits) to achieve a beautiful musical interpretation. This happens less in slow waltz and quickstep.
In quickstep, apart from the faster tempo, beats 1 and 3 are equally accented. So you can start a figure on either beat 1 or 3 and it won't really feel any different. Beat 1 can still be identified by the start of a new phrasing, or theme (often by a return to the 'home' major chord also)but it's a little more difficult than in foxtrot.
Why tango is specified as 2/4 - i don't know. I always count it as 4 beats to the bar.
Here's a useful link, but it's a bit technical:
http://doubleU doubleU doubleU.musictheory.halifax.ns.ca/13ts.html
(sorry about the double Us but if i typed the W's it wouldn't post).