Here's Mrs Thatcher's Death cert as an example of an English certificate. That is the only official notice of death (Wills and newspaper obituaries may have more info)
http://www.theguardian.com/politics/2013/apr/16/margaret-thatcher-death-certificate
and an earlier example here https://www.gro.gov.uk/gro/content/certificates/images/Death%20Cert%20PDF.pdf
For England and Wales, to find a death cert from 1837 onwards , go to http://www.freebmd.org.uk/cgi/search.pl and enter details. Almost all of the index entries are now in this database.
You will find an index reference number which looks like this 'March 1954 2a 222'
When you have obtained the number go to
https://www.gro.gov.uk/gro/content/certificates/most_customers_want_to_know.asp#CertificateInformation
Don't use a non government site like Ancestry because they charge a lot more. At the moment they cost £9.25.
Hi Kairen
You can order death certificates from Ireland via this web site:
http://www.irishgenealogy.ie/en/civil-records/help/i-want-to-get-a-copy-of-a-certificate-what-do-i-do
the site also has many links to Irish records including earlier wills.