The reason that salts are used is that people tend to choose the same passwords, and not at all randomly. Many used passwords out there are short real words, to make it easy to remember, but this also enables for an attack.
As you may know, passwords are generally not stored in cleartext, but rather hashed. If you are unsure of the purpose of a hash-function, please read up on that first.
Now, what the attackers can do is to simply generate a list of common passwords and their corresponding hashes. Comparing the hashes that a site has stored with the table will, if common passwords are being used, reveal the passwords to the attacker.
A salt is simply added to make a common password uncommon. A salt value is generated at random and can be fairly small, the only purpose is to lower the probability that the hash-value will be found in any precalculated table. A common way to combine the salt and the password is to simply concatenate them, i.e. the stored hash value is Hash(salt||password). The common password "password1" now magically becomes, e.g., "6$dK,3password1" and is very unlikely to be found in a table.
The salt can be stored completely in the clear in the database, next to the hashed value. Once the attacker has the database and wants to find the passwords, he needs to generate the precalculated table for each salt individually, a costly operation.
Yet another way to defend against this kind of attack is to slow down the attacker. This can be achived by iterating the hash-function many times, i.e. storing Hash(Hash(Hash(Hash.....(Hash(salt||password)))...). Also, a pepper can be used, which is another random value concatenated to the password, such that the stored value is Hash(pepper||salt||password). The pepper is then not stored at all and therefore all possible values of the pepper need to be tested when trying to log in. Using 8 bits for the pepper give 256 possible values, which is very fast when the true user tries to log in. However the attack will work 256 times slower since all pepper values need to be tested for each password!