It seems that my iPhone's (4s - purchased Nov. 2011) battery loses its charge fast.
I have read all the relevant OPs. Is there any app which provides information about the battery's health? Or should I simply change it?
Actually there is a little trick that might work for you.
If you go to Settings > Privacy > Diagnostic & Usage (before iOS 8, it was in Settings > General > About > Diagnostic & Usage > Diagnostic & Usage Data), then you'll see all of the logs generated by iOS; usually for crashes.
On this page, you MIGHT be able to see a log starting with 'LowBatteryLog...'. If you select it, you'll be able to see some additional information about your battery.
I checked this on my iPad before posting this and the log wasn't listing the capacity of the battery but I strongly remember that when I checked this on my iPhone before, I saw my iPhone's battery's current capacity in mAh which was a very useful information.
If you happen to see a log related to your battery, this Wikipedia article about iPhone 4S clearly specifies it's battery.
iPhone 4S Battery: 3.7 V, 5.3 Wh (1,430 mAh)
Besides this, I don't think there is another option for battery health check besides jailbreaking and installing some apps outside Apple's App Store.
Before taking any action related to your battery, you can check iMore's guide on saving battery life. They're not going to supercharge your battery life or you might already be familiar with them but it's always worth a look.
Another option could be buying a case with a battery in it. I strongly suggest Mophie's iPhone 4/4S section for you if you're not thinking about changing your iPhone soon.
If all else fails, and you really want to change your battery in the end, you may take it to an Apple Store and have an expert deal with it but if you're familiar with some basic tech. fixing, you can even do it yourself via iFixit's iPhone 4S battery change guide since changing iPhone 4S's battery is a very simple process. iFixit even sell parts and tools for the job.
the current os does not seem to show this data or at least not as described above. This is an excellent app which tracks battery health for both the laptop and iPhone.
There's also an app designed to test your battery:
The score it gives isn't the easiest to interpret, but you can compare it against other scores.