You'll likely already know about the big apps like Facebook, Twitter or Whatsapp so I won't talk about those, but here are a few apps that are lesser known (or at least not that big, which I like).
Yahoo Weather
You talked about a weather app. The Yahoo Weather app is really slick. The standard iOS weather app is actually based on the Yahoo one.
Google Maps
Yes, this is not a small, unknown app. But I still want to mention it: (where I live), it's got the most accurate and up to date apps. Additionally, it's got a lot of real-time data. It's not as beautiful as Apple's app, though.
Shazam
You've also likely heard about this one. If you hear a song and you want to know what it's called and who's listening, Shazam is all you need.
Banking / financials
My local bank has a great app for checking my current financial situation. I can easily transfer money and check if my employer has already paid my salary. For me, probably one of my more useful apps. Check if there's an app for your bank.
Postal / delivery apps (FedEx, UPS, etc)
There are some great delivery apps out there which give realtime information about your package. If you shop online, these apps can be real handy.
My favourite app: Spotify
I really like Spotify. Everywhere I go, I can listen to any song I want. Some people are a bit hesitant about these subscription services, but if you listen to music regularly, than it's worth it.
I'd really recommend you to try out Spotify, or any other music subscription service. It's awesome.