What does the N designation mean in organic chemistry? For example: N-hydroxymethylurea?

It just emphasizes how the substituent is attached to a nitrogen. That's all! :) Since urea is an amide, it follows amine/amide naming conventions that specify when a substituent is bound onto a nitrogen and not bound onto something else.

You probably meant to say N-(hydroxymethyl)urea.

Another name for it is methylolurea; ol nicely emphasizes that it's an alcohol group attached to urea in some way, and methyl emphasizes that it's not a #"CH"_3"OH"# (with no other bond on #"C"#), but a #-"CH"_2"OH"#.

Also, be careful that you don't confuse hydroxymethyl with methoxy (I sometimes do). Hydroxymethyl implies an alcohol substituent, but methoxy is an alkoxide (deprotonated alcohol) substituent.

N-(hydroxymethyl)urea says that a #-"CH"_2"OH"# group is attached to a nitrogen on urea, #"H"_2"N"("C"="O")"NH"_2#, in place of one of the hydrogens. The structure is like this:

As an aside, if you wrote:

N,N'-dimethylurea

Then it shows that the first nitrogen emphasized (N) is not the same as the second nitrogen emphasized (N'), and each singular methyl is on a different nitrogen.