5th, 6th, 1st, 3rd - Name

[Sped Tiger]

Could you tell me, folks, what we call these numeral endings in English? (Are they called accretions?)
E.g.: 1st, 2nd, 3rd, 4th, 1278th.

 

[5jj]

They are the ordinal suffixes.

 

[Skrej]

I realize you seem to be asking about the affix specifically, but I'll add this as a little extra information. While these are ordinal numbers (and thus have ordinal suffixes as 5jj stated), the other type of number is referred to as cardinal.

Cardinal numbers tell us how many (five, twelve, ninety).
Ordinal numbers tell us the order (first, third, tenth).

One easy way to remember the difference between the two is that 'ordinal' sounds a bit like 'order'. They actually do share a common Latin root.