Did they favour a particular side? Did they sell weapons to either force? Did they exploit the divided nation in any way? Or just sit back, and watch the show?
Like most other powers in Europe, Britain recognized the Confederacy as a belligerent party. This permitted their ships to dock in British ports, for them to trade with British subjects, but it was precious little more than that. It certainly did not extend to any sort of official alliance, though the Confederacy sought repeatedly but unsuccessfully to draw them into such an arrangement, initially by denying invaluable cotton to British industry. The closest it ever came to happening was probably in late 1861, when a US Navy ship stopped a British steamer to remove two Confederate diplomats en route to England. The crisis was defused diplomatically, however, and Britain receded firmly into neutrality again. There was further talk in 1862, following Lee's early successes, to approach the Americans in concert with the French (who were more pro-Confederate anyway) and suggest an armistice and separation. Growing Union success put an end to this sort of talk.
Beyond the official level, British subjects, operating in a variety of business ventures (gunmakers, cannon founders, shipbuilders, textile concerns, shoemakers, etc) made huge contributions to the Confederacy's ability to sustain itself. They were paid, generally, in bales of cotton smuggled through the blockade by the same fast, stealthy steamers that brought the British goods in. A regular trade developed, with ships dropping off cotton in Nassau or Bermuda, picking up coal, luxury items, and military goods, and then making the run into whatever unoccupied Confederate port was easiest to reach. Late in the war, this became almost exclusively Wilmington, NC, a wartime boom town through which flowed hundreds of thousands of rifle-muskets, uniforms, shoes, blankets, cloth, leather accouterments, cannon, cartridges and shells, et al. Absolutely huge profits were made (the cotton was both valuable and cheap), down to the sailors involved; a half-dozen bottles of rum or gin, bought for a dollar each in Nassau, might fetch ten dollars a piece in Wilmington.
Additionally, commerce raiders were effectively bought from British shipyards, on the condition that they had to be armed after delivery. So a ship would be sent to whatever neutral port the Confederate agent arranged, a captain and a few officers would be smuggled in from the Confederacy, and a crew built using foreign soldiers sailors. They would arm the vessel, provision it, and go out to wreck Union commerce.