Yeah, I've been getting this same problem.
Lots of times, my iPhone does a hard reset on its own before being able to install and will not install successfully. So, I have lots of experience with this.
What you need to do is, connect the iPhone to iTunes.
Then, in the menu bar at the top of the screen, their is an option to "Check for purchases" or something like that (sorry, I am at work and not in front of iTunes... just look for something like that.) I think it is in the advanced menu.
iTunes should then download the application and install onto the iPhone when you sync.
If you get an error about the application not being able to install, then you need to click on your iPhone in the devices section of iTunes and deselect the application from syncing. Sync to fully delete the application.
Then, select the application to sync and you should have no problems.
Unfortunately, you may lose saved data depending on the application.
It has been a long time since I have experienced this many problems with any of Apple's products. I just don't think they are keeping up with the growth of their company and product line, and as a result, quality standards have certainly declined. My iPhone was great until 2.0 came out. Hopefully, they will get back on track.
Let me know if this worked. If you are having trouble following my steps, I will repost when I get home later today and I am in front of iTunes to tell you exactly where and what to click.