Don't think you're missing any parts Bob, ;-)))))))

The answers you got, pretty well cover the whole gamut of possible
problems you could have.
My first thought would be you have a morse taper for a drill press,
and those have a tang (flattened end) on them and this will not fit in
a lathe, you will have to cut or grind the tang off, or return and
trade for a morse taper without a tang.
Next possibility is that your tail stock will eject your morse taper
when retracted all the way back (a good thing to have), in which case
you only have to crank the barrel out a little and then your morse
taper should fit, the way to install a morse taper is by inserting it
with a swift flick of the wrist.
And yes all mating surfaces should be clean, that counts for tapers,
spurs and chucks etc.
Smaller MT ??? well that should be obvious, as there's a large size
difference, would not fit anywhere close.

Have fun and take care
Leo Van Der Loo

What part of this am I missing?

BobMac