Bede is referring to the discrepancy in the dating of Easter between the Irish/"Celtic" church and the Roman church. Until about the seventh century, those dates sometimes overlapped, but often differed because they were calculated according to different formulas. This was particularly problematic in England because, while they had substantial contact with the continental church, the island had largely been evangelized by Irish monks, who brought with them the Irish computus (the term for the Easter formula).

The Irish don't seem to have been terribly concerned with the discrepancy between the calculations. They largely considered it a matter of ritual rather than belief. Since ritual could differ from place to place without any serious theological ramifications, it wasn't a huge issue. For the Roman church, however, the dating of Easter was of far greater importance. Many considered it a matter of belief and therefore attempted to impose the Roman formula on the rest of the church.

It's worth noting that Ireland was not the only region with a divergent computus. There hadn't been a lot of standardization to this point, and various Eastern churches also differed from Rome in their calculations.

It's also particularly important to note that the attempted imposition of the Roman computus wasn't necessarily coming from Rome itself. There were Irish ecclesiastics who argued for reform from within the Irish church, and it seems to have become a divisive issue. Rome itself seems to have taken a backseat to the whole Easter controversy -- although Columbanus wrote to Gregory the Great, there's no evidence that he ever replied. On the whole, the papacy seems to have been primarily concerned with the unity of the church rather than with which was the correct formula. Within the Irish church, Cummian was perhaps the most ardent proponent of the Roman formula. He too was concerned with Christian unity, and since most of the western church adhered to the Roman formula, Cummian argued that the onus was on the Irish church to conform.

Although it originated within the Irish church, the matter actually came to a head in Northumbria with the Synod of Whitby in 664. There, King Oswy ruled that the Northumbrian church would follow the Roman formula, citing the primacy of Rome as the see of St. Peter. From Northumbria, it spread throughout the island and into Ireland (where some churches had already accepted the Roman formula). The Synod of Whitby didn't end the controversy immediately, and it took about another century for all the insular churches to conform.

For more information on the early Irish church and the Easter computus, here are a few resources:

Kathleen Hughes, The Church in Early Irish Society -- a good overview, but dated. Hughes tends to overemphasize the extent to which the Irish church was segregated from the continent.

Dáibhí Ó Cróinín, ed. A New History of Ireland, vol. 1 -- the section on the church that deals with the Easter controversy is also written by Hughes, but it's a more moderate perspective. There are lots of other good essays in here as well about insular society.

For a more specialized perspective:

Immo Warntjes and Dáibhí Ó Cróinín, eds. Computus and its cultural context, 300-1200 -- this is a collection of essays from a conference on the computus.

Several articles by Dáibhí Ó Cróinín (see his bibliography) - he's pretty much the established expert on the Irish computus at the moment