London Bridge is the oldest bridge over the River Thames. While the current incarnation of the bridge dates from the 1970s, there has been a bridge in this place since around 50 AD, when the Romans drove some wooden piles into the river's mud. Since then there has always been a bridge here, and for a long time it was the only one. (Nowadays there are many bridges crisscrossing the Thames.)
Sadly, London Bridge is not one of the prettiest of the Thames bridges, although its name might be the most famous. Expecting the name to conjure up something special, people often mistakenly call Tower Bridge London Bridge. This leads to the story that an American bought London Bridge in 1968, thinking he'd bought Tower Bridge: what he did buy now spans a lake in Arizona.