When I try to make a Battle.net account, I'm asked to put in my NID. I have no idea what it means, and so don't know what to type in.
First, Battle.net is divided into two completely separate and distinct partitions, known as Worldwide and China, each of which has completely separate accounts and game servers. There is no connection whatsoever between the two partitions. An account on one partition does not exist on the other.
A Battle.net account exists within a partition.
Within each partition are regions. The regions are game-specific and so different games may use different regions. An account may play the same game in multiple regions, but will have different characters, achievements, etc., in each region, which do not transfer between regions.
The China partition has only one region, CN. In the Worldwide partition, there are multiple regions:
An account may authenticate to any region in the partition, but still play games in a different region in the same partition. The only benefit to authenticating to a different region is the possibility of reducing the latency of the login process.
With all that out of the way:
In China, Blizzard (actually, the operator NetEase) is required to ask for players' National ID numbers. Everyone in mainland China has such a number, so it is mandatory to provide one if you register an account in the China partition. (However, the last time I checked, Blizzard doesn't actually verify these numbers with the government, nor does it check the validity of the ID.) Due to the unique state of Taiwan's political situation, it does not use the same system of NID as China. The same is true about Hong Kong and Macau, where both places have separate ID systems. As such, it is optional to provide the NID if you register an account in the Worldwide partition.
As Jiao has said, NID means "National ID" and really only applies to Taiwan. Also, I've found that you can just leave it blank with no problem.