Hi Cheryl,
Thank you for your message.
Have you tried looking in Church records for information on the family? Most Catholic records are held locally so you may need to write to the local parish priest for possible assistance. One website that you may find useful is the Irish Times where they give an overview of what records are available in specific parishes. For Taughboyne/All-Saints parish, follow this link:
http://www.irishtimes.com/ancestor/fuses/rcparishmaps/index.cfm?fuseaction=showidrecords&CityCounty=Donegal&parish=All-Saints&churchid=1003
For Church of Ireland records check http://ireland.anglican.org/about/42 and http://www.nationalarchives.ie/.
www.familsearch.org also have some Irish birth and marraige records in their database
You could also check for record of the family in the land acts pertaining to Ireland in the 19th century. There are two:
The Tithe Applotment Books (1823-1838) found at http://titheapplotmentbooks.nationalarchives.ie/search/tab/home.jsp
and Griffith?s Valuation (1848-1864) found at http://www.askaboutireland.ie/griffith-valuation/
Civil registration began in Ireland in 1864, so you may be able to find some records of the family in these records which are available from the General Register?s Office in Dublin. Here is their website: http://www.groireland.ie/research.htm
You can search the indexes to this records online at https://familysearch.org/search/collection/1408347
For more information go to http://www.irish-genealogy-toolkit.com/Irish-civil-registration.html
The latest records that are available online are the 1901/1911 census records. You could try searching in these to see if there are any Andersons still in the area at the time:
http://www.census.nationalarchives.ie/search/
I found some earlier records of Andersons in Donegal online also, however there is no genealogical information in these so it is difficult to know whether they are your Andersons or not, but you may be interested anyways.
The first is on the FLax Growers List. This was part of a government scheme to encourage the linen trade, free spinning wheels or looms were granted to individuals planting a certain area of land with flax. The lists of those entitled to the awards, covering almost 60,000 individuals, were published in 1796, and record only the names of the individuals and the civil parish in which they lived. There is a reocrd of a James Anderson in Taughboyne parish:
http://www.failteromhat.com/flax/donegal.htm
Also, Donegal Genealogy Resources have the indexes to the record of wills and deeds from the Dioecese of Raphoe 1684-1858. There are a few references to Andersons:
http://freepages.genealogy.rootsweb.ancestry.com/~donegal/wills.htm
Some other websites that you may find useful during your research are:
The National Archives of Ireland http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/
The National Library of Ireland http://www.nli.ie/en/family-history-introduction.aspx
The National Archives UK ? genealogy search: http://www.nationalarchives.gov.uk/records/looking-for-person/
The Public Records Office of Northern Ireland http://www.proni.gov.uk/index/family_history.htm
Genealogy Links: http://www.genealogylinks.net/uk/ireland/donegal/index.html
I hope that this is helpful. Remember to post as much information as you can with regard to the people you are researching. The more information you post, the more likely it is that one of our volunteers will be able to advise or assist you. Also include information concerning which sources you may have already used so others may further your search.
Kind regards,
Genealogy Support