One of the sayings in library-land is “give'um what they want”. Usually, this means popular fiction, DVDs, music, free computer time, wifi, and upcoming technology such e-books. In terms of RPGs, giving your potential players what they want is usually the same thing as giving them the most popular games and the equipment needed to play them.
In 2008, rpg.net asked what the most popular RPG was. Although they didn't actually define what “the most popular” actually meant (How often people play? Most sales on Amazon? Liked the most?) the results were both consistent and inconsistent.
In every opinion presented, except for one, Dungeons and Dragons was rated the top RPG. This makes sense for several reasons. First, they were the first RPG to be created and marketed. Second, they seem to have a loyal fan-base who is welling to purchase their books, adventures, and minis. Third, the name “DnD” seemed to combine all of that particular game's incarnations – everything from Advanced Dungeons and Dragons to v3.5 to the latest version.
Dungeons and Dragons has definite pros and cons. A pro is the brand name – many people at least recognize the name Dungeons and Dragons. In fact, DnD has celeberty power as well as many novels, several movies and a television show from the 1980s. The rules can be as simple or as complex as one wishes, depending on the whim of the Game Master. It can be played by children as young as 12, but if you want to include even younger children in your games, there is a rules-light version for children as young as six.
Unfortunately, DnD can be more about combat and dice rolls than about acting and story telling. Some players may like this, while other players may find it annoying. I have heard it said that because of how the powers of version 4.0 have been created, Dungeons and Dragons now resembles a table-top game of World of Warcraft. The skills are greatly downplayed, the character's roles are similar, and even the races are similar. (Because I do not play WoW I am making this comparison based on the experiences of several friends.) If this is true, this quality may appeal to some players and drive other players to distraction. A large number of maps, miniatures and dice are “required” to play Dungeons and Dragons. Home-made versions of miniatures using pieces of cardboard could be substituted, as can graph paper for maps. However, if one decides to purchase official minis and map tiles, lost pieces may become an issue. Finally, Dungeons and Dragons has had some very bad press, particularly from religious objectors who see it's violence, use of magic, demonic artwork, and fictional pantheon of gods as “spiritually dangerous”. Related, there is also (an unfounded) fear that fantasy role-playing, as personified in Dungeons and Dragons, leads to mental instability. If you choose to run a game of Dungeons and Dragons at your library, you may need to deal with these negative connotations.
Dungeons and Dragons is a system that has stood the test of time. With the amount of new books that are being published, it still has a loyal fan-base, and seems to be attracting new players. Although only the Player's Handbook (and maybe the Dungeon Master's Guide... and the Monster Manual...) is required to play DnD 4th edition, supplementary books* contain new options for players and game masters alike. A traditional high-fantasy dungeon crawl using one of the oldest RPGs may be the way to kick off your RPG program.
*These supplementary books, the Player's Handbook II and III, include character classes that were iconic in previous editions of DnD, such as the monk, the druid and the bard. Please note, in the 4th edition, the bard doesn't suck. This is a good thing!