Why in plain TeX it says "New counters are allocated starting with 23, 24, etc. Other registers are allocated starting with 10"? I want to know what's the difference between counters and registers in TeX. If anyone could enlighten me on this that will be highly appreciated.
First of all counters in plainTeX are registers!
New counters are allocated starting with 23, because the lower register numbers are already in use. From the documentation of plainTeX:
% The following counters are reserved: % 0 to 9 page numbering % 10 count allocation % 11 dimen allocation % 12 skip allocation % 13 muskip allocation % 14 box allocation % 15 toks allocation % 16 read file allocation % 17 write file allocation % 18 math family allocation % 19 language allocation % 20 insert allocation % 21 the most recently allocated number % 22 constant -1
So most of the reserved counter registers are needed to manage the allocation of registers. And that is the reason why other types of registers does not need such a lot of reserved register. For other registers with ten reserved registers those reserved registers are for free temporary usage.