In recent history (mid 19th century on), nothing in particular. Some might be politicians or businessmen, but many are nothing more than rich private citizens. There have been a number of privileges that peers have had over time (called, you guessed it, the privilege of peerage), but most of those are gone now. Since 1999, most earls don't even have a seat in the House of Lords anymore.

Retired "commoner" PMs were often given earldoms as a reward for their service. Harold Macmillan was the last to accept one of these titles.

The main "rights" held by earls, include the right to be addressed as "The Right Honourable The Earl [of] X", the right to wear a specific dress at formal event, the right to be tried by other peers if prosecuted, the right to use a specific crown on their coat of arms, and the right to have a private audience with the queen (which no one uses because no mechanism exists to schedule such meetings).

Quick note before we start: The 1707 Act of Union consolidated England and Scotland to form Britain.

An English peerage would have entitled a man to sit in the House of Lords. But after the Act of Union, only sixteen spots were allowed in the Lords for Scottish peers, who were chosen to represent Scotland in the British parliament. Some peers held important positions within the British ministry. For example, eighteenth-century First Lords of the Treasury(what we would now call Prime Minister) included the Duke of Newcastle, the Earl of Bute, the Earl of Chatham, the Earl of Shelburne, etc. Peers might also be members of the royal household (e.g. Groom of the Stole). Others held posts called sinecures—jobs that provided significant income without much work (e.g. Lord Privy Seal). Another option would be to act as a diplomat (e.g. the Duke of Bedford served as the British Plenipotentiary during the 1762 negotiations for the Treaty of Paris). Those within the monarch’s household and ministry would have been present at his levées (official court receptions), and less well-placed peers might have attended to speak with the king.

Several eighteenth century peers felt compelled to write memoirs or diaries of time spent in government or character sketches of important men they knew. More personal pursuits would have included answering correspondence, managing household accounts, traveling, riding, and perhaps hunting. Genteel people of the eighteenth century had leisure time to spend on hobbies, such as botany, history, geology, etc. The specialization of professions that would not occur until the nineteenth century. Educated men (who we would today see as amateurs) could contribute to the advancement of fields of knowledge. Of course for many hobbyists these were true passions, but it was also advised that people have hobbies so that they could carry on interesting conversations with other well-rounded people.

Gentility in this age was strongly associated with domesticity. British lords would have spent many evenings with their families, but when they went out, they might attend a dinner party or the theatre. (Of course the frequency of how often a man would have gone out in the evening would have depended on the season, where he lived, his age, and his social circles.)