Blocks

Quickmail--allows email and attachments between student and student or student and teacher
Course Menu/Course Menu +--gives access to administration links as well as a navigation system, all in one. Course Menu + even has chapters! This allows us to use one block to do the job of a couple, and it lets us collapse the topics so that students do not have to scroll so much for a large course with much content.
My Courses--allows collapsible view of categories and course, and editable view as well. Works well if you set up multiple semesters/years on your server and students or teachers are overwhelmed by a long list of past courses.
Attendance--this one hasn't been updated for 1.8 yet, but my faculty liked it very much for Moodle 1.6.

MRBS4Moodle - This is a block that I maintain that we use to allow teachers to schedule our computer labs. It provides a single sign on so teachers do not have to login again. Students can lookup and see if there class is scheduled in a lab but they cannot make reservations.

Feedback - The feedback module (more so than the block) allows for creating surveys like course evaluations. It was nice because with the template, it was easy to give the rest of the faculty the same questions for their course evaluations which allowed for things to be standardized (but also with the ability to customize after they pulled in the template). Hopefully it will get included into Moodle core soon (if not 1.9 perhaps 2.0).

Gradebookplus_v2 - While technically not a block, it provided greater flexibility for inputing grades. Hopefully much of this functionality will be included with Moodle 1.9 making it obsolete (at least that is the hope).

I have been impressed with how the best blocks end up making their way to Moodle core.

Feedback and Quickmail. They both fill a very basic need missing in Moodle...at least up through version 1.6

The feedback module is by far the best 3rd party module out there...if any 3rd party module should be in core, this is it...in my opinion.

Steve

Best Module: Survey

Best Block: Course Menu

Both provide fantastic functionality, are well supported and well used by my teachers.

Quite a while since this original question, but this might be useful...

We (HRDNZ, Moodle Parter) have set-up a site which is just used for testing of non-standard modules and blocks...
(and other 'bits and pieces' of code or modifications).

http://www.elearning.school.nz

We will try and develop this to include as many non-standard blocks and modules as we can
(time permitting) - so others can at least see them in action.

Feel free to create a login and experiment
(please note there are no guarantees of high-availability of this site as it can break at any time).

Stuart