HDR stands for High Dynamic Range.
Your camera's sensor can only capture a certain range of brightness from the darks to the lights. This is the dynamic range. Cameras these days are around 12-14 stops of light between black and white.
Some of the scenes you may try to photograph contain a brightness range bigger than what your camera can capture. To work around this, you can take multiple photos of the same scene using the bracketing settings, some underexposed to capture the brights and some overexposed to capture the darks. Then you feed all of those images to the HDR software and it creates a single image that now contains all of the brights and darks beyond what your camera could capture in just one image.
The old school alternative is to do the same exposure bracketing and just use the one image you like the best.
So,, when I`m usingg my Iphone to take photos and it makes two types - the simple photo and the photo in HDR format, better to take the HDR? It takes more memory. Also, if I`m taking photos to instagram does the HDR format influense the quality of the phot when I`m applicating filters?
HDR is not a format, it is a process.
You might be confusing HDR with raw files or DNG files?