Fingal is sort of a county and sort of not. It's an administrative area of County Dublin, but it has its own county council and signs on its borders saying "Welcome to County Fingal".
The island of Ireland is known for having 32 counties, 26 in the Republic and 6 in Northern Ireland. However, roughly one third of the Republic's population lives in one county, Dublin, because of the disproportionate size of Dublin City, the nation's capital. The city itself is looked after by Dublin City Council, but the surrounding county came under the auspices of Dublin County Council. As the area was so heavily populated, however, this body had a great deal more responsibility than any other single council in the country.
In 1994, therefore, Dublin County Council was split into three administrative areas, Fingal (covering North County Dublin), South Dublin, and Dun Laoghaire/Rathdown, covering the area south-east of the city. All responsibilities of the old county council were devolved to these three bodies, giving them the same status as any other county council in the country. This is why the areas are often referred to as counties in their own right. However, County Dublin remains intact as one of Ireland's 32 counties, consisting of the three new administrative areas, and the city itself.
Here's some info from the Fingal County Council web site (http://www.fingalcoco.ie/):
The administrative area of Fingal is located to the north of the City of Dublin and extends in the west to the
line of the River Liffey, to the east to the sea and to the north as far as the county boundary. It covers an
area of approximately 173 square miles and has a population of almost 168,000 making it the 5th. largest
local authority area (on population) in the country. It came into being on the 1st January, 1994 following
the abolition of the former Dublin County Council and the establishment of 3 new authorities to replace it.
As far as I can make out, the name Fingal was the name given to Fionn Mac Cumhail, the hero of Irish Mythology, by the 18th century scottish writer James Macpherson. In 1763, Macpherson wrote Fingal: an ancient epic poem in six books, which was loosely based on the Ossianic Cycle. Not only did he decide that Fionn wasn't a good enough name for his hero, he also decided that Fionn, Oisín and Oscar weren't Irish after all, but were in fact Caledonian.
I'm not sure how this version of the name came to be associated with North Dublin, but it probably has something to do with the locality of Finglas, which is a neighbourhood on the north side of Dublin City.