A few comments and some suggestions:
Songs Vary: some songs just have a steady beat, others have a pronounced pattern that will make the difference of feeling on or off. Depending on the dance, being off may feel fine until you will miss each major transition such as speed or intensity (being off by half a cycle, or worse by a few beats).
Examples of what to look for:
*A waltz with a pronounced downbeat on 1 & softer 2-2-3: ONE!-2-3, 2-2-3 -VS- no perceptible difference: 1-2-3, 1-2-3, 1-2-3...
*a foxtrot/swing vs a lindy: does the music have a cue for the quick step? slow-slow-quick-quick vs lindy 1 2 3&4, 5 6 7&8 (-VS-steady 1 2 3 4 5 6 7 8 with no indication)
**Good practice song: 1234 by Feist (it even counts for you!)**
*where does the music transition: If there is a dramatic change, the music may warn it is coming, but the change should actually occur on the 1 (usually).
Other tips:
*Listen & Count Out Loud! rather than dance: don't try to do too much. (Yes, the follows should typically have an easier time than someone trying to plan a next step or more)
*(non-waltz) Start bobbing or tapping to the steady part of the beat (I use my foot), then add in the count by feeling (i add in emphasis with an air guitar or drumming, then use that to find the 1).
*foxtrot/swing: sometimes Slow-Slow-Quick-Quick (or rock-step) is easier, other times 1-2,3-4,5,6 is easier.
*AVOID Dubstep while honing this skill: it plays with sound and is likely to do tricky things, for example lindy to Sail...
*In dance, don't let yourself drop steps while doing a move. Keep the count with your feet.
*Worry about the 1's, half-way points, the last step, or quick-steps (what do the dance steps naturally emphasize?). The evens and odds are way too much to try to track.
*listen for the quick-quick or step-AND-step (which means either the start or the half-way is next!)
*listen to a song, find the count, and WALK around moving in any direction to the count. Break out of the standard dance form and learn to instinctively step to the count wherever you move.
*Unless it is for competition, dancing is social. Don't be so focused on succeeding or failing the music's count that you break YOUR dance. (sometimes the music is even a hindrance)
Story: I have danced to a song with a periodic halt that my partner and I missed each time and then did when the music withheld the halt and it was a lot of fun for both of us listening, trying and totally bombing it.
*final recommendation: get a group of friends, put a song on with a recognizable count.
Get on count and start dancing and doing moves. Focus on checking that you are on right part of the count.
If you get off, correct it (don't stop and restart, rather, correct while still dancing).
If you are off and don't seem to realize it or correct it soon, your friends standing around watching should be listening to the music and watching you. They start saying the count out loud ("one two three-And-four five six seven-and-eight") until you have corrected and seem to be stable on count.
Start easy and go more difficult.
(optional) After a little while, perhaps have a friend start bumping into you while you are dancing (not your partner). You must keep the count even while they attempt moderately to interrupt your normal step pattern.
*this does 2 things: helps you hear the count, and helps you fix it amid a dance.
