Sweeney Todd is a Victorian horror character and urban legend. He was a barber who murdered his wealthy customers and robbed them before sending the corpses to his partner in crime, Mrs. Lovett, a baker who made them into the filling of her meat pies.
While the story has a fine history in "penny dreadfuls" and other media, in fanfiction there is really only one Sweeney Todd, and that is the Sweeney Todd portrayed in the musical film Sweeney Todd: The Demon Barber of Fleet Street (2007), the titular character of which is played by that perennial fangirl-magnet, Johnny Depp.
For further details of the legend in its various incarnations, visit the Real World Wiki.
In Canon Edit
The Sweeney Todd film is actually an adaptation of a Broadway stage musical by Stephen Sondheim, and as with most theatre, characters can be slightly different depending on how each actor portrays them. However, there are some important traits that are always the case. In canon, Sweeney Todd is convinced that everyone deserves to die, obsessed with killing everyone, and in love with his razors. (Seriously, he sings a love song to them.) Mrs. Lovett will do anything to attract Sweeney and/or make a profit. It is canon that they sleep with each other at some point—Mrs. Lovett has a line about "me rumpled bedding legitimized" when fantasizing about marrying Sweeney—but Sweeney is still more loyal to his wife Lucy, who poisoned herself.
In Fanon Edit
A very common and, as mentioned above, mostly canon pairing is Sweeney/Lovett.
In Badfic Edit
There are mainly two types of badfic: fics where a Mary Sue waltzes in and wins over Sweeney and fics where a Sued-up version of Mrs. Lovett does the same thing. Almost all badfic tends to portray Sweeney as a woobie, ignoring the "insane, cannibalistic serial killer" part.