Welcome to Ireland Reaching Out and apologies for the delay in replying to your message.
As you can see from the link below, Dromara is a Civil Parish in Co. Down:
http://www.irishtimes.com/ancestor/fuses/townlands/index.cfm?fuseaction=TownlandsInCivil&civilparishid=820&civilparish=Dromara&citycounty=Down
Church of Ireland records for the parish are lost while records for the corresponding Roman Catholic Parish of Dromara, part Ballynahinch begin in 1844.
There are some places where you can search online, however these websites do not have complete collections and many do not cover the period after 1900:
www.familysearch.org has a huge database of genealogical records including some church records for Ireland.
www.rootsireland.ie have a large collection of Irish Church records, however you have to pay to use this site.
Other important sources include the Tithe Applotment Books and Griffith?s Valuation. Even though these sources only name the head of household, the information is still important:
Tithe Applotment Books (1823-1837): http://titheapplotmentbooks.nationalarchives.ie/search/tab/index.jsp
Griffith?s Valuation (1847-1864): http://www.askaboutireland.ie/griffith-valuation/
Some other websites that may be helpful are:
The National Archives of Ireland: http://www.census.nationalarchives.ie/search/ you can search the 1901/1911 Census on the National Archives page too, for instances of the family name
General Register Office Northern Ireland: http://www.nidirect.gov.uk/family-history
Passenger Lists: http://www.castlegarden.org/
Remember to post any new information that you find here. 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.
Best of luck with your search
Clare Doyle
Genealogy Support