Probably in order of usefulness:
-Clock (it is a watch!)
-Messages/Alerts (to check who's texting or calling without pulling phone out)
-Wallet (mainly for boarding passes)
-Starbucks (for mobile pay)
-One password (quick jump to common used passwords)
I would probably use news apps and instagram more if they weren't so slow... hopefully the push for native apps will fix this.
I'm guessing you're mostly asking about 3rd party apps. The only one that I've found useful on a daily basis is HeartWatch. If you're at all interested in tracking your heart rate this is certainly a huge winner. It tracks resting HR, workout HR, waking HR, and sleeping HR. It works pretty well as a sleep tracker too.
WUnderground is also the weather app that I use. I find it to have a much better complication than the default. I've heard good things about Carrot, but WUnderground is free!
Those are the most frequent ones I use. The watch isn't really the greatest platform for apps.
For 3rd party, PCalc Lite (EXCELLENT btw, added one .99 package otherwise free), Bring! (Like it for shopping so I don't have to walk around with my phone out while going down the shopping list--once left my phone in the cart and left the store...yikes that was scary; so glad someone was honest and turned it in!), and occasionally TuneIn Radio to control the station choice and play of my frequent radio stations while my iPhone is out on the countertop or desk for example and I'm doing things around the house. These are the only 3rd party apps I'm using on the watch now. I know PCalc is native, think Bring! is too. Off hand unsure of the other.
As for Apple apps I use the most: Heartbeat, battery, calendar, activity, weather are in my Glances. As for accessing as stored apps, I use Timer, Alarm (both are used as complications on certain watch faces too) and Messages..all the time. Same for Phone. And I do use the Friends Side Button all the time for contacting people. Love that it immediately gives me a phone or message choice for a person. And on phone choices I frequently have several listed numbers for each person and they are all just right there. Depending on the ambient noise at the time Siri may not always be the best choice for asking to make a call or to send a quick message. As long as my hand is free I find this a super fast way to access the contact. I get mail on my watch.
Wink (turn lights on/off shortcuts), Workouts, the heartbeat monitor, sometimes the music control (like when mowing the lawn and my iPhone is in my pocket), Wallet, Apple Pay (no, not really an app), and also Siri to start navigation or playing certain songs.
Notifications almost first. I am in many situations like meetings or other things where pulling out my phone is inappropriate. the AW allows me to see incoming messages and emails discretely and that is positive.
Boarding passes for plane trips. One of my favorite features is using my AW to check in when boarding a plane (I travel a lot for work)