I don't know much about this topic, so I'll refer you to a wikipedia article; http://en.wikipedia.org/wiki/Cosmic_string
. Perhaps as a non-expert I can help explain part of the story.
It is believed that there were various dynamical fields fluctuating about when the universe was much hotter, right after the big bang. As the universe cooled, these froze, much as the atoms in a liquid freeze into a regular crystal pattern as it cools. When atoms freeze, they can accidentally get stuck in an imperfect pattern, for example one where there's an extra sheet of atoms in one region. The edge of that extra sheet is a one-dimensional defect, called an edge dislocation. (http://en.wikipedia.org/wiki/Dislocation
has some pictures.) There's extra energy per atom in that defect, because the atoms there cannot join the low-energy crystal pattern. The energy is comparable to the binding energy between the atoms, which is also comparable to kT, where T is the absolute melting temperature and k is Boltzmann's constant.
For the hypothetical frozen fields, there also would be some large energy scale, corresponding to the temperature at which the fields froze out to form our present vacuum. As for what the specific thickness of these strings would be and what sort of fields would be involved, there are many different ideas, discussed in that article. All the ideas involve a large energy scale and a short distance scale, so all involve a lot of energy per unit length. Lower energy scales and longer length scales are ruled out because we already know about what fields are relevant there, from experiment.
(published on 07/06/2011)