On the numbers, different authors come up with differing estimates, but I think what follows is a broadly-accepted quantitative depiction.

For western and central Europe (excluding Hungary, the Balkans and Russia but including the UK), Jan de Vries (European urbanization 1500-1800, Methuen 1984) puts the population of towns exceeding 10,000 inhabitants at 12.2 million in 1800 (10% of the area's total population), 29.6m (16.7%) in 1850 and 66.9m (29%) in 1890. But from your dates I imagine you mean continental Europe only. So minus the British Isles his figures come to 9.7m (9.1%) in 1800, 20.7m (13.8%) in 1850 and 46m (23.9%) in 1890, consistent with a rise from 12% urban in 1830 to just over 30% in 1913.

Within de Vries's area, the fastest growth outside Britain occured as we might expect in Germany, Switzerland, Austria-Bohemia and Poland which saw rapid industrial acceleration from a low base, but also perhaps more surprisingly in Scandinavia. Urban growth of course wasn't confined to this area: while only about half as urbanised as western-central Europe, Russia, Hungary, Greece and Bulgaria also saw rapid increases in urban population after the mid-19th century, though data here are less reliable for the early part of the period.

Unsurprisingly given the rise of the proportion urban, most of the increase came from migration from country to town. That doesn't mean towns were the death-traps sometimes imagined: by the 19th century the early modern excess of urban deaths over births was eroding. Landers indeed indicates (Death and the metropolis, Cambridge 1993) that London may have experienced natural increase from the 1780s, though migration contributed the majority of growth until at least the 1830s. De Vries reports a similar reversal in Berlin around the 1820s, in Amsterdam around the same time and in Stockholm from the 1870s.

De Vries and Norman Pounds (An historical geography of Europe 1800-1914, Cambridge 1983) both note the wide area from which migrants were drawn: Pounds singles out movement from eastern to western Germany, along with smaller Italian and Belgian movements into nearby parts of France, though a substantial part of the last seems likely to involve agricultural labour. Agrarian distress from the late 1840s and depressed agricultural prices from the 1870s added to the flight from the land, although emigration to the Americas and other regions of more recent European settlement also drew off tens of millions.

Spatially, the industrialisation of the period reinforced the concentration of Europe’s economic and urban centre of gravity to the northwest, a trend already visible for centuries but more pronounced from the 1870s, with industrial England, the Low countries, northwestern Germany and northeastern France constituting the relatively highly urbanised region that persists to this day. Milan represented the centre of a secondary but older region extending from the Rhone to the Adriatic, and Naples of a smaller less heavily industrial area including Rome and Palermo.

One noteworthy feature of the urbanisation process is that entirely new foundations were very few among the rising industrial towns of the period: almost every large city even among the fastest-growing was a place of significance by the 18th century, most tracing their urban origins to the Middle Ages or even earlier. Capitals were among the fastest-growing cities, and ports were also prominent, reflecting the growth of government, finance and maritime trade alongside industry.