How do we assess whether or not mathematical concepts have been grasped and understood?

Preparatory questions and links:

Focusing questions:
What concepts are easy to grasp and understand?
What role does formative assessment play?
How do you assess understanding?
Can you, do you or should you teach towards a particular assessment?
Do you use different forms of assessment for a concept or group of concepts? If so, what ones?
Should explanations by students be written or spoken or both? Should they use 'mathematical' language?
What concepts do you think are difficult to assess? Are they also difficult to teach?
Do you always tell students how or when they have been assessed? If so, how: grade, can-do, %...? If not, why?

(Provided by @ColinTGraham)

Threaded discussion from Thursday, 12th August 2010 -
Threaded discussion from Monday, 16th August 2010 -

Links mentioned in the discussions, with contributor:
Northwest Evaluation Association provided by @ghewgley
Some Math Assessment Sites provided by @cybraryman1

Resources (costing money!) mentioned in the discussions, with contributor:
Interactive Mathematics Program suggested by @CoachGinsburg

Other links which may be useful
Pupils do better at school if teachers are not fixated on test results suggested by @ColinTGraham
Institute of Education press release about their study on focusing on results suggested by @ColinTGraham