I have 478 MB available, and this game is 228 MB. Every time I try to download it, I get a message saying I don't have enough space. But I do. What am I supposed to do?! I really want this app!
You actually don't have the required space.
Apps are shipped in containers called IPAs. An IPA is actually just a zip file, and compresses some of the content. Each app, depending on how they are built, can achieve varying degrees of compression.
In the above image, you can see just how much disparity exists between the App Store's reported size and the actual install size.
Infinity Blade 3 is listed at 1.7GB by the App Store, but installs to 2.6GB. Additionally, Bastion reports requiring 534MB, but actually needs more than twice that amount of space (1.2GB).
Since it would be impossible for Apple to gather the required information from actual install size, they offer the size of the IPA as a guideline. To be safe, one should allow for 3x the existing space. In your case, you may require about 800MB free (ballpark).
And remember, you need some free space (for caches, app data, etc.). It's unwise to operate with absolutely no free space, typically leading to a negative impact on performance.
Bear in mind that for an App to be installed you need more free space in your device than the actual App's size. This is so because the download is done in a temporary folder (you download an .ipa file, which is compressed). Once the download is completed, the App is uncompressed and installed in the appropriated directory.
This is true for updates, too. When an update is downloaded, it is installed in a new App directory, and then the user's data is moved from the old installation directories into the new ones. The old installation is then deleted.
Whenever you download a file, it's not the size you think it is. It may say 200 MB but turns out to be 600 MB. For example when you download a rar, you unpack it and the rar is only 2 Gb but when you unpacked it, it's like 4GB.
This happens because the RAR is compressed, simply the app format is compressed but when you have it, it's larger in size.
The best way to free up space is to go to Settings > General > Usage and to look at whats taking up your storage. You'll probably find apps that take up a lot of storage that you never even use! This is probably the best way to free up space.