1. Government classification
The purpose of classification is ostensibly to protect information from being used to damage or endanger national security.
Classification formalizes what constitutes a "state secret"
Different levels of protection based on the expected damage the information might cause in the wrong hands.

2. Classification levels
(from the highest level to lowest):
Top Secret (TS) 
The highest level of classification of material on a national level. Such material would cause "exceptionally grave damage" to national security if publicly available.
Secret
Such material would cause "serious damage" to national security if publicly available.
Confidential
Such material would cause "damage" or be "prejudicial" to national security if publicly available.
Restricted
Such material would cause "undesirable effects" if publicly available. Some countries do not have such a classification.
Unclassified
Technically not a classification level, but is used for government documents that do not have a classification listed above. Such documents can be viewed by those without security clearance.

3. Classification levels
Depending on the level of classification there are different rules controlling the level of clearance needed to view such information, and how it must be stored, transmitted, and destroyed.
Additionally, access is restricted on a "need to know" basis. Simply possessing a clearance does not automatically authorize the individual to view all material classified at that level or below that level.
The individual must present a legitimate "need to know" in addition to the proper level of clearance.

4. Compartmented information
Often there are additional constraints on access, such as (in the U.S.)
Special Intelligence (SI) which protects intelligence sources and methods,
No Foreign dissemination (NOFORN) which restricts dissemination to U.S. nationals, and
Originator Controlled dissemination (ORCON), which ensures that the originator can track possessors of the information.
Documents in some compartments are marked with specific "code words" in addition to the classification level.
“Code word" classification can be more restricted than top secret.

5. Atomic information
Government information about nuclear weapons such as nuclear warheads often has an additional marking to show it contains such information.
"Extremely Sensitive Information" is applied.

6. Sharing classified information between countries
generally employ a special classification scheme which both parties have previously agreed to honor, for example:
COSMIC TOP SECRET (CTS),
FOCAL TOP SECRET (FTS),
NATO SECRET (NS),
NATO CONFIDENTIAL (NC), and
NATO RESTRICTED (NR).
NATO UNCLASSIFIED (NU) information.
This is NATO property and must not be made public without NATO permission.