Why won't things download in my AppStore on my iPhone 4?

I have been having a really hard time with downloading things onto my iPhone 4 from the AppStore. When I click on the "free" button and then click it again to install it sits for a couple seconds and then goes back to the regular screen with the "free" button. This has been happening for a couple weeks now. Also, I keep getting pop ups that want me to put in my email password and it's weird. I've gone into the settings and redid all my information for iCloud and AppStore and I can't find the problem.

I'd restore your phone with iTunes.
Or just restore The appstore on ITunes when you plug in your phone.
Best of luck to you!

Wireless Data Speeds with different iPhones:
iPhone 4 = 3G
iPhone 4s = 4G
iPhone 5 = 4G-LTE

I have the iPhone 5, which operates on 4G-LTE, which is (up to) 10x faster than 3G. But when I try to download Apps while I'm in a "3G area", Apps sometimes do have trouble downloading, and occasionally will stop.

Your iPhone 4 operates on 3G which is a bit slower, plus if you have less "service bars" (connection) the wireless data is even slower.

You should try and download or even Update Apps when you have these conditions to optimize your success:
1. Download Apps when you are in a Wi-Fi area, and have a Wi-Fi connection (should be much faster than the 3G connection).
2. Download Apps when you have a Full Signal Strength your iPhone 4, don't download them if the Signal Strength is only 1, 2, or 3 bars out of 5 total.

These steps might help.

Also: You might test your download/upload speeds on your iPhone 4 at different locations:
The site called Speedtest.net has a simple "Mobile App" (see Link below) to test and see download/upload speeds and this could help you notice if your speeds are slow, before trying to download Apps.
1. Use the "Mobile Test App" when at home on Wi-Fi - write this down (try several times)
2. Use the "Mobile Test App" while at home but NOT on Wi-Fi (cell data speeds / 3G) - write down - (try several times)
3. Use the "Mobile Test App" when you are away from home, but before downloading "iPhone Apps"; see if the speeds you are getting are okay first, before trying to get an "iPhone App" to download. If the speeds are slow, wait until you are somewhere else and have a better speed.

Note: For me with the iPhone 5 on 4G-LTE good speeds are about 5 to 7 Mbps (Mega bits per second).

Well… I don't really have enough information to give you an exact answer but ill give you a few things that might help.

Try to log onto iTunes on your computer and see if you can download things that way.
go to appleid.apple.com and check and see if it shows any activity that you don't recognize. Also, I'm not sure how long you have had the Apple ID but in the past you could create an Apple ID without it being an email address. If you have one like that there's a thing on this site that says "Make your primary email and your apple Id one and the same" Do that because there have been a few times where that fixed similar issues

For security purposes sometimes your devices will ask you for that sort of information just to verify who you are… It does it randomly, I honestly don't understand why it does, but it does.

I wish I had a direct line that I could give you to call but I don't have one so your best bet would be to just call 1-800-My Apple and tell the call director that you are having issues with the iTunes app on your phone recognizing your apple Id and they will transfer you to an iOS Advisor. You won't have to pay a service for this issue because Apple ID support is always free.

But most importantly, if it keeps doing it don't sit there and try to mess with it much because its not very hard to accidentally end up putting your phone into a DS lock, and that is just one giant headache.

But yea, just call support and they will be able to get everything working for you pretty quickly

May be your phone needs an update or restore through. It may be a bug as you just said the pop up arrives too many times. Just do a restore on your phone. Hope that serves good.