This is the storage where the OS, your downloaded apps and messages reside. While companies advertise the space provided as 8/16/32 GB, only a fraction of it will be available for end usage as most of it is taken up by the OS. Having low free internal storage will lead to your phone slowing down and most of its features will end up not working. Android and iOS do not support installation of apps on external storage media so if you're a person who downloads a lot of applications then buy a phone with a lot of internal storage.
External Storage can supplement the internal storage of your phone. Photos, music, videos and data can be stored on an external MicroSD/SDHC/SDXC Card when the internal storage gets too cramped. Not all phones have a card slot or support all capacities.
Universal Flash Storage (UFS) is a common flash storage specification for digital cameras, mobile phones and consumer electronic devices. This could bring higher data transfer speed and increased reliability in flash memory storage, thus reducing market confusion and the varying number of adapters for different number of cards. It's currently used in the Samsung Galaxy S6 and is 2.7 times faster than the Galaxy S5’s memory and tests actually revealed a 3.25x performance increase.