Coming from a Windows background, the only compressed file format I'm used to is DDS, which uses S3 DXT compression.
I'm unsure if DDS is supported on iOS devices, and what other options there might be? I've heard of things like Ericsson Texture Compression (ETC) and PVRTC but I don't understand how they all fit together and how much they rely on specific hardware functionality.
All iOS devices support PVRTC. Unfortunately it produces noticable artifacts when compressing sprite-type graphics and is restriced to power of two square textures. The new PVRTC2 format produces much better quality and removes the size restrictions. Unfortunately, Apple decided to disable this feature even though their hardware supports it. The other alternative is ASTC, which has similar quality as PVRTC2, but this is only available to the newer A8 devices using OpenGL ES 3.0.