Certain other institutions in Westeros do have elections, but they are not true democracies. The ruling council of the Faith of the Seven, the Most Devout, elects new High Septons - but only the Most Devout are allowed to vote, and new members of the Most Devout are appointed by High Septons. Similarly, the Order of Maesters elects each Grand Maester, but only the ruling council of Archmaesters known as the Conclave are allowed to vote, and new Archmaesters are appointed by the Conclave itself. The ancient Kings of the Iron Islands were actually non-hereditary elected leaders, though only major lords and ship captains could vote. The exact process that the wildlings use for selecting a new King-Beyond-the-Wall is unclear, but it appears to be more of a general acclamation by different tribal chiefs, not an actual voting system.
In contrast, the Night's Watch is a true, representative democracy: every member of the Night's Watch, from the First Ranger and the other high-ranking officers, down to the cooks, and even the lowliest steward who cleans out the chamber pots, has an equal vote when electing a new Lord Commander.
Lord Commanders serve for life, so a new Choosing is only held when the current Lord Commander dies. In the interim between his death and the Choosing, one of the other high ranking officers will serve as acting Lord Commander, but it is frowned upon for this to continue for an extended period of time.
Night's Watch Choosing for the 998th Lord Commander of the Night's WatchIn the year 302 AC, Following the death of Lord Commander Jeor Mormont during the Great Ranging beyond the Wall, along with many other high-ranking officers, the Watch was left in a precarious position. Ser Alliser Thorne stepped up to serve as acting Lord Commander during the emergency, until the threat of Mance Rayder's wildling horde was surprisingly halted in the Battle of Castle Black. The victory left the Watch with enough breathing room to formally hold a Choosing for a new Lord Commander. The major candidates were Ser Alliser and Ser Denys Mallister, commander of the Shadow Tower.
At the choosing, Samwell Tarly put Jon Snow's name forward as a candidate, after learning that he will not accept Stannis's offer of legitimization. Sam reminded all of the black brothers present that Jon was Jeor Mormont's personal steward, that he led the force to deal with the mutineers at Craster's Keep, and took command of Castle Black's garrison during the battle when Thorne was wounded. As a result, Jon was chosen as the new Lord Commander of the Night's Watch.
In the books
In the A Song of Ice and Fire novels, the election of a new Lord Commander of the Night's Watch actually requires a two-thirds majority to win. The TV series apparently simplified this to a simple majority: in Season 5 it is only referred to as "a majority", and visually just two equal stacks of vote-tokens are displayed (so Aemon can dramatically cast the final vote which breaks the tie). This allowed the TV version to quickly condense the somewhat more convoluted political wrangling which enabled Jon Snow to achieve a two-thirds majority in the book version.
If a two-thirds majority is not reached, voting will continue the next day, with no time limit. In the many centuries that the Night's Watch has existed, there was once a choosing that lasted for nearly two years, including seven hundred votes - though the voting process typically only lasts several days, sometimes just a few hours. This may be loosely inspired by real-life papal conclaves, in which voting goes on as long as it has to until a majority is reached: typically papal elections only last a few days, sometimes only hours, but once in the thirteenth century a papal election indeed lasted over two years (1268-1271) (though unlike a papal conclave, everyone in the organization of the Night's Watch gets a vote).
The Night's Watch only had about 10,000 members during the Targaryen Conquest (though it may have had even more in past millennia), but the Targaryen unification of the Seven Kingdoms caused the Night's Watch to drastically decline. With the number of internal wars drastically reduced, the number of new recruits from the losing sides in those wars dried up. By the start of the War of the Five Kings, the Watch had less than 1,000 men - thus the Watch never experienced great difficulties in counting votes from each man in the Watch. Actual voting takes place at the headquarters of the Watch - which for many millennia was the Nightfort, but over two hundred years ago it was abandoned and the headquarters was moved to Castle Black. Not all men can be physically present for voting, because they have to guard their posts at other castles on the Wall. Instead, the commander of each castle on the Wall travels to Castle Black, and is trusted to accurately report the tally of votes from his castle.
Any man of the Night's Watch may nominate himself or one of his sworn brothers to stand in the election, though usually the number of candidates will quickly be reduced as the election focuses on a few men who have a large number of supporters. Major candidates tend to be major officers or commanders of castles on the Wall, with leadership experience, but even black brothers who held no major offices have been elected Lord Commander, usually if they had performed exemplary service. Lord Commanders usually tend to be elected from the Rangers, though this is not always the case. Sometimes if the officers see that a new recruit has leadership potential, they will make him a Steward - so he can serve as a personal Steward to one of the officers or the Lord Commander himself, to actively groom him for command. This is perceived as fast-tracking a recruit for command or even to stand for election as Lord Commander some day.
Given that Jeor Mormont was the 997th Lord Commander, there have in theory been 997 Choosings - though within the narrative Samwell points out that they don't know if there have actually been 997 Lord Commanders, stretching back a full eight thousand years, as the early millennia of the Watch are shrouded in legends. Nonetheless reliably recorded Choosings stretch back for many centuries before the Targaryen conquest three hundred years ago.
In the fifth novel, during an argument about Jon's decision to send Val (Mance Rayder's sister-in-law) on a mission, he tells his subordinates that if Val does not return "then you may have a chance to choose a lord commander more to your liking". It is unclear if by this he meant to imply that there is a process for removing a Lord Commander, or if he was alluding that he didn't expect to live very long if his plan failed.