Over the course of the 20th century, physicists painstakingly cobbled together a reasonably efficient "standard model" of physics. The model they came up with almost works, without resorting to extra dimensions. It merges electromagnetism with the strong and weak nuclear forces (at almost impossibly high temperatures), despite the differences in their respective strengths, and provides a neat theoretical framework for the big, noisy "family" of subatomic particles.
But there is a gaping hole. The standard model doesn't include the gravitational force. That's why Jove, the physicist in Jeanette Winterson's novel, Gut Symmetries, calls the Standard Model the "Flying Tarpaulin" - it's "big, ugly, useful, covers what you want and ignores gravity." Superstring theory aims to plug that hole.
According to string theorists, there are the three full-sized spatial dimensions we experience every day, one dimension of time, and six extra dimensions crumpled up at the Planck scale like itty-bitty wads of paper. As tiny as these dimensions are, strings - the most fundamental unit in nature, vibrating down at the Planck scale - are even smaller.