Halo seems to stick to a formula that many find enjoyable. It deviates somewhat from the set norm in some areas, but plays it safe as a series. The first Halo started off a new set standardm with the prospect of being limited to 2 weapons, regenerating health, and the control scheme. It was more adaptable for consoles than most shooters around at the time, and felt easier to pick up. Some could argue that it casualised the genre somewhat.
These days most shooters seem to mimic this set standard, even the popular Call of Duty series. Before, shooters allowed players to carry multiple weapons, and made use of health bars. Nowadays most use the 2-weapon regenerating health system, even PC shooters.
Before Halo, FPS territory was mostly PC based (bar some exceptions such as Goldeneye and Perfect Dark), and console itterations were somewhat overlooked.
Halo makes changes to its system in some subtle ways, Halo 2 bringing online to the series, and increasing the popularity of console online gaming, as well as introducing dual wielding, Halo 3 bringing items such as bubble shields, changeable match types, forge, Halo 3 ODST didn't really add much apart from firefight, which mimiced Gears of War's horde mode, Halo Reach took a step back somewha by introducing the class system, though through these classes they brought new abilities such as the jetpack.
The differences between the Halo series and other game series are subtle, some of the 'innovations' have been brought to us by other series, it could be argued that it's a generic series, but its impact on the genre, even if it's just through its sheer popularity, is rather abundant.