Hi, I live in Sweden.
BankID is using certificates of some kind (a cryptographic key stored
in a file or on a "smart card", which can be your bank card or ID
card).
These are issued by banks, as an example. You use them with a code.
There are card readers for home computers so that you can log on to
the bank, and other things as well, with them and your password. You
can choose between that solution and a file for the certificate.
You can use a card with your BankID anywhere that it's supported. It's
mostly banks and goverment agencies that you can use ot for
identification with (but you rarely need to use a BankID specially
rather then your normal ID, it's just when you need to use a
certificate).
Some text (probably somewhat badly) translated from swedish on that
site:
"The customer identifies himself in your services by showing his
electronic ID and supplying his security code. A check of the validity
for the customers BankID is done automatically in the validation
software (BICS) that is required in your services. Then the electronic
ID is verified against a current blacklist. If the electronic ID is
valid and the correct security code is supplied you know that the
right person are allowed to get access to the information."
"The customer signs or accepts documents in your services the same way
as when logging in and identifing himself. A check of the validity for
the customers BankID is done automatically in the validation software
(BICS) that is required for tou services. [...]"
I could have gotten something wrong there or be missing something
since I haven't used it myself, but that information should be
accurate.