The personnummer is a ten digit Number given to all Swedish residents so that they can be identified by the system. It is defined by paragraph 18 of folkbokföringslagen (1991:481). The official translation is 'Personal Identification Number'; a rather strange translation in my opinion. It is unique to every person and is used in all sorts of databases as a unique key (or whatever it's called), efficiently locking everyone without a number out of the system.

Some compare the personnummer to the American social security number, but I think that is a somewhat bad comparison since the personnummer is given to all residents but the SSN only to those who qualify for social security. (I might be wrong here)

It looks like this: 123456-7890

The first six digits are the date of birth with the year first, then month and date last. Digits 7 and 8 do not have any meaning. (Although one of them used to give place of birth.) The ninth digit gives the sex of the person, odd for men and even for women. The last digit is a checksum calculated according to the modulus-10 method. The dash is just there to separate the date of birth from the other digits and can be omitted. It is replaced with a plus (+) sign if the person is more than 100 years old.

Since the plus sign isn't stored in databases this used to create some confusion. For example, people above 100 years of age weren't allowed to withdraw money from their bank accounts (the computer system thought they were too young). This is the reason to why the numbers are stored with 12 digits (with four digits for year of birth) in most computer systems nowadays.

The number system first appeared in its current form in 1947, but then without a checksum. The checksum was added in 1967.
Many other countries, such as Russia, China and Finland use a similar system. The numbers given to organisations (juridical persons) have the same format but the digits have different meanings and the checksums do not match. It's quite useful for databases though.