As far as I could research on this subject, much of it's present expresions are based on the so called Early Medieval art of Britain and Ireland. Art historians typically begin to talk about "Celtic art" only from the La Tène period (broadly 5th to 1st centuries BC) onwards.
In this case, I rather prefer to employ the "Insular Art" label also known as "Hiberno-Saxon" art.
But does celtic art is sheer celtic?... Certainly not.
Masterpieces like the Book of Kells, Book of Durrow, pocket gospels like the Book of Mulling, Book of Deer, Book of Dimma, and the smallest of all, the Stonyhurst Gospel (now British Library), plus insular decorations and designs like spirals, triskeles, knotworks, carvings in stone, especially stone crosses, circles and other geometric motifs, are perhaps the most related examples of evoking celtic art.
The interlace patterns that are often regarded as typical of "Celtic art" were in fact introduced to Insular art from the Mediterranean, both directly and via the animal Style II of Germanic Migration Period art, though they were taken up with great skill and enthusiasm by Celtic artists in metalwork and illuminated manuscripts.
Most Insular art originates from the Irish monasticism of Celtic Christianity, or metalwork for the secular elite, and the period begins around 600 AD with the combining of 'Celtic' styles and Anglo-Saxon (English) styles ('zoomorphic interlace' decoration as found at Sutton Hoo).
The finest period of the style was brought to an end by the disruption to monastic centres and aristocratic life of the Viking raids which began in earnest in the late 8th century. These are presumed to have interrupted work on the Book of Kells, and no later Gospel books are as heavily or finely illuminated as the masterpieces of the 8th century.
But perhaps the most relevant matter is that the influence of insular art affected all subsequent European medieval art, especially in the decorative elements of Romanesque and Gothic manuscripts.
It began as a conscious effort mostly in the British Isles, to express self-identification and nationalism, and became popular well beyond the Celtic nations, and whose style is still current in various popular forms, from Celtic cross funerary monuments to interlace tattoos.
Coinciding with the beginnings of a coherent archaeological understanding of the earlier periods, the style self-consciously used motifs closely copied from works of the earlier periods, more often the Insular than the Iron Age. Another influence was that of late La Tène "vegetal" art on the Art Nouveau movement.
The late 19th century reintroduction of monumental Celtic crosses for graves and other memorials has arguably been the most enduring aspect of the revival, and one that has spread well outside areas and populations with a specific Celtic heritage. Interlace typically features on these, and has also been used as a style of architectural decoration, especially in America around 1900, by architects such as Louis Sullivan, and in stained glass by Thomas A. O'Shaughnessy, both based in Chicago, with its large Irish-American population. The "plastic style" of early Celtic art was one of the elements feeding into Art Nouveau decorative style, very consciously so in the work of designers like the Manxman Archibald Knox, who did much work for Liberty & Co..
Interlace, which is still seen as a "Celtic" form of decoration, somewhat ignoring its Germanic origins and equally prominent place in Anglo-Saxon and Scandinavian medieval art, has remained a motif in many forms of popular design, especially in Celtic countries, and above all Ireland, where it remains a national style signature.
The Secret of Kells is an animated feature film of 2009 set during the creation of the Book of Kells which makes much use of Insular design.
As Stephen Walker comment in Dalriada Magazine , "a few Celtic motifs have meanings that are more-or-less a consensus of contemporary Celtic designers and artists. The meanings attached to these symbols can often be traced to the rediscovery of Ireland's cultural history in Victorian times as well as the emerging sense of national identity in Scotland, Wales, Isle of Man, Cornwall and Brittany as these cultures struggled to maintain their unique traditions and characteristics".
The current Renaissance of Celtic Art adds new agendas and a new imagination about how the old relates to the new. Some recent Celtic symbolism is very innovative adaptations of contemporary concerns, intellectual fashions and spiritual trends. Artists are by their nature creative and imaginative. An intuitive sense of symbolism should be recognized for what it is; the communicative intent of the artist.