Biblically there are three domains. Private, semi-public, and public.
Private domains would be like a house, or a fenced yard. (Apartment buildings require an eruv chatzerus...a different subject)
The parameters for Public domain are probably not what you think. Generally, it has to be a busy market place. The definition of "busy" is a subject of debate, but the generally accepted definition is that 600,000 people (the number of adult males at the time of the giving of the Torah) have to regularly pass through, and that the area is never closed off to civilians. Examples given in the Talmud are roman street markets.
Using those parameters to define "public" (obviously there is more to it, but for simplicity sake, we'll go with just that) there are a lot of areas that fall between private and public. According to the strict biblical law, carrying in a semi-private domain, like a suburban side street, is ok. The rabbis decided that carrying in a semi-private domain might lead to carrying in a public domain. Anyway, in order to mitigate the risk of carrying from a semi-private domain into a public domain, the rabbis decided to forbid carrying in a semi-public domain, unless there is an eruv. The discussion about why and what is mostly found in Gemara eruv in, although it's hard to learn practical law from it. Remember, that biblically, carrying in a semi-private domain is ok. So the rabbis did not permit something that was forbidden. They forbid something that was permitted, and then made conditions for exceptions.
Now, what's with the strings, you might ask. The simple answer is that in order for a semi-public domain to be considered "like private," it must be walled off. Like a walled city. But even walked cities have gateways. So the rabbis permitted doorways and gateways. What about a wall made completely of doorways? Yes, that's ok. What is the absolute minimum that you can call something a proper doorway? At least two poles with a string attached. And that's where the idea of an eruv being a string around an area comes from. But it's not as simple as four posts with a string connecting them. The land also has to be private. Usually the Jewish community will "lease" the land in the eruv from the city for a few dollars. There are strict rules about how far each pole can be from the next one to still be considered a doorpost, and strict rules about how the string has to be attached to be called a lintel.
