A good way of thinking about it is as a clockwork watch or clockwork mechanism or machine. You could have two watches keeping the same time but with different time signatures. Think of the second hand as the pulse with its tick, tock, tick, tock and the beat as the mechanism driving the second hand as 123, 123, 123, 123. Each tick or tock is supported by three beats. A beat behind the second hand pulse of 1234, 1234, 1234, 1234 would be a 16/4 time signature. A beat that is the same as the pulse 1234, 1234, e.t.c. would be in common time. An old mantelpiece clock could be imagined in cut time 2/2.
That's why trying to determine the time signature of a tune by counting foot taps as beats only works in 4/4 or if you double or half your tap interval to an 8/4 or 2/4, e.t.c. Complex time signatures can't be determined that way. For example, if you tap out a 6/8 jig, you'd fool yourself into thinking it's in 2/8 time, i.e. two beats to every bar whereas you are actually tapping out groups of three beats, 123, 123.
A common question in traditional music that might help is the difference between a jig and a reel. To real (forgive the pun) traditional musicians, who have learned by ear and never seen a musical score, the reply will be something using words; if you can say 'jiggery, jiggery,' to the music, it's a jig and if you can say 'this is how a reel goes, this is how a reel goes,' it's a reel. The jiggery is Ji Ge Ry, 123, beat on the capital letter, and this is how a reel goes is Thisis Howa Reel Goes, 1234. Therefore, a jig is in 6/8 time (usually) 123, 123, i.e. 6 X 8th notes (beats) per bar and a reel is in 4/4 time 4 X 4 notes (beats) per bar.
To understand the 123 grouping think of a waltz which is often (usually in 3/4 time, i.e. three quarter notes per bar. You'll find yourself saying 123, 123, 123, 123 while tapping out the pulse as one tap on the 1 of each 123 grouping.