There are lots of good apps for teaching mathematics to children but I would like to learn about apps for undergraduate/graduate/research levels.
Helper questions
Any algebra system (like Maple, Sage)?
Interactive geometry (like GeoGebra, Cabri, etc)?
What else?
On
Wolfram has a whole series of apps designed for undergrad/grad students. Check this link out: http://products.wolframalpha.com/courseassistants/
On
Scientific calculator, say TouchCalc http://appshopper.com/link/touchcalc
Graph drawer, say GraphCalc http://appshopper.com/link/graphcalc
Logic Games, say Cat Physics http://appshopper.com/link/cat-physics
On
I'll answer the part about computer algebra systems. I have iCAS and PocketCAS Pro, but my experience is too short for detailed reviews.
I'm not sure any of these apps would be of great use on iPhone or iPod touch because of their small screens but you may try. I used iPad mini.
This is a port of REDUCE and should be the most advanced CAS for iOS. However, it's interface is tricky to master and the absence of arrow keys on the keyboard makes life really hard. But it is accompanied with tutorials and you can always contact the author. Currently I'm studying the ways to have a better keyboard there. So the most advanced iOS CAS comes with the steepest learning curve.
It also comes with GNUPlot for graphics.
This is my primary iOS CAS at the moment. If you are just looking for a single CAS to start with go with PocketCAS Pro. It's CAS part is (based on) Giac, nothing to add here I think. Its keyboard is much better then the one of iCAS but still entering expressions is at first quite tedious.
I haven't tried that one but it is probably the most popular CAS on the AppStore. It utilizes its own CAS engine referred to as MobileCAS.
Yes, that's true, Texas Instruments has come with their TI-Nspire CAS software properly ported to iPad. However TI calculators are aimed more to be an educational tool and their CAS capabilities are quite poor compared to other mobile apps and desktop systems. Meanwhile this app is the most expensive one (30 dollars!). We might hope that for that price you will recieve solid quality of TI products, but I can't judge this claim due to my limited experience with the app.
Like PocketCAS Pro its CAS engine is Giac, but it features interface that mimics the look of TI-92/Voyager 200. You may think that it is a stupid idea too emulate TI calculator interface including monochrome LCD display on a far superior hardware, but in fact it's not. After having lots of troubles with apple's ios keyboard I was looking forward for such an app. Unfortunately CAS Calc P11 is really buggy as of January 8 2013 and I do not recommend it.
WolframAlpha is awesome, just remember that it's not a CAS, but a reference application. If you forgot what is $e^{i \varphi}$ it comes in handy.
As it was mentioned there is a SAGE app, but I can say nothing about it, I'm not on good terms with SAGE.
The other possibility I haven't studied yet is using your iOS device as a client to your computer running desktop CAS. I believe the most promising approach is to use shell clients to reach the console interface of your favorite CAS. Personally I would choose Axiom/FriCAS, but Maxima is also an option.
I'll update the answer as I gain more experience.
Since you mentioned Sage, I will say that you can use Sage online for free at http://www.sagenb.org/. So, it can be used on any device that can connect to the internet.