Crucial functionality. My favorite feature of Tiger.
Every time I use my computer I use the dashboard widgets, and in the course of a day, I invoke them dozens and dozens of times.
The reason: speed, simplicity, ubiquity.
Speed: There is NOTHING faster. One button access. F12 - boom, I got 'em. Or one squeeze of the mouse. This is faster than any app which I'd have to get from the app folder on my dock, and also faster than selecting the app icon on the dock (with a button or mouse squeeze I don't need to aim or search). Plus, unlike an app, it launches instantly - all my widgets are immediately ready to use.
Simplicity: there are many more choices of widgets than dedicated apps, and often they emphasize simplicity instead of bloat - a dedicated widget to just one function... perfect. Plus it is so simple to evoke it!
Ubiquity: all widgets are omnipresent, instantly available, but out of the way if I don't need them. Perfect.
How I use them: my number one use is notes. I use two widgets for that, one is K-notes and the other is NotePad. One note is just a list of bills I have to pay every month with dates due. As I pay bills, I check off "paid", and that way can see what I already paid and what still needs to be paid and when. Nice way to keep track of bills due. If I need to jot down a note really quick, a widget note is perfect: simple, fast, and always there the second I need it, but not in the way. I also keep reminders there, my contacts list and similar things I may need frequently.
I also use the calculator widget from PEMDAS:
http://www.apple.com/downloads/dashboard/calculate_convert/pemdas.html
I also use daily: Apple's conversion widget, weather, dictionary, calendar, and iStat. And other widgets I use very frequently, like a shipment notification widget (I buy a lot of stuff online), some I use more rarely, like flight info.
Dashboard and widgets are a huge, huge productivity booster. This is truly an enhancement of functionality.
So for me, the appearance of widgets was a very, very big deal. I cannot use a computer without them. Absolutely critical.