As my needs change and as apps improve, the list of programs I run on my many Android devices (8 devices, at last count) changes. Here are the ones I find most useful, and think you might like them too:
Best WiFi. When your home or office has multiple WiFi access points, Android devices don't switch to the strongest signal automatically. This is what Best WiFi does.
(The limitation to this app is that all access points need to have different SSIDs. If your location has access points that use the same name, then there is another app that will do the switching. Sorry, I forgot its name, but it is out there!)
WatchDog. I find that the Android devices I use a lot (phone, tablet) start to slow down. They become painfully slow after waking up, and sometimes it can take minutes to get back to normal speed. WatchDog is a dynamic task closer that shuts down apps operating in the background that consume too many CPU cycles. Since installing this app, I have not had to wait any longer.
AquaMail. I don't like GMail's manner of forcing emails into threads, because I find it is too easy to miss the most recent email. On my phone, I now use AquaMail, because it eschews threads: all emails are listed in the order in which they arrive, as God meant them to be.
CheckItOff. I need reminders, but I don't want to plug up Calendar with trivial reminders, like the next doctor visit. CheckItOff is a widget that lists tasks under each day. I love it, because it has the best interface I have found for entering tasks. One flaw is that it does not display the due time, so I have to add it manually.
DicePlayer. There are all kinds of video players for Android, and Dice was the first to enable hardware playback. Over the years it has improved with useful functions like a bar showing how far through each video is, press the spacebar on tablets to pause and play, drag the screen to change the brightness and volume.
GasBuddy. Tells me where the cheapest gas is near me, all over Canada and USA. Before long trips, I use its map mode to see which stations are cheapest along my route.
Railteam Mobile. Gives me information for nearly all trains in Europe. It was especially helpful on my trip earlier this month when a delayed Deutsche Bahn train made me miss the day's last connection in Frankfurt.
Widgetsoid. This app lets me customize the information displayed by status bar. For my phone, I set up Brightness (10%, 30%, 100%, and Auto), WiFi toggle, Mobile Data toggle, and Orientation toggle.
WorldClock. One of the least fancy clock apps, but I like this one because it creates a scrollable widget. This means that I can see the weather for the home towns of me (in Canada) and my daughter (in Germany) in the space of a single widget.
PhoneUsage. A widget that reports the number of texts, minutes of phoning, and megabytes of data I have consumed in my monthly plan. It lets me know when I am getting close to the limit, so that I can switch to TextPlus or Skype.
Of course, there are other apps I use, but these are some of the most important ones.
In summary, I find it amusing to read reports from longtime iPhone users who get exposed to Android, and are stunned by the functionality they are missing. The list I present here may well end up making iPhone diehards envious.