This is in effect is the same as a national insurance number in the UK. All non-residents who buy a property and therefore need to have connections with the tax system are required by law to have an N.I.E.number.
You will need an N.I.E. number for all kinds of purposes; the solicitor who handles the sale will apply for this on your behalf as part of the service.
If you wish to work in Tenerife providing that you have your N.I.E. and your Residencia, you will not require a work permit.
This document has to be applied for by individuals who intend to reside in the country for more than 183 days per year. You apply for this via the local police station.