Since attributes are really just built-in skills that everyone has, everything about attribute checks is the same as for skill checks (as described in Skill Checks in the Introduction). Also, like skills, whenever a check is made against attribute, and you roll exactly what was required for success (the effective attribute level), the attribute is permanently increased by one (see also Improving Skills in Skills).
How do you know when an attribute check is required? Basically, whenever your character tries to do something which is not covered by a particular skill, it will require a check against the appropriate attribute. Also, certain other rules (notably those in the Weight and Encumbrance section of Other Rules) indicate that attribute checks must be made occasionally.
There are no hard-and-fast rules about when to make attribute checks, and what difficulties and modifiers may be appropriate. There are some examples scattered through the rulebook where appropriate, but it is generally in the hands of the GM (although players frequently make suggestions).
Some examples may help to clarify things.