Brief Hydra overview:
1. The Illuminati: Grant Ward's faction, provisionally the Illuminati - was born from a Hydra faction in the US that backed the US during WWII. Howard Stark was an associate and close ally at worst. A rival Hydra faction assassinated him to deny the Illuminati the super soldier serum. A secondary goal was preventing Howard Stark from entering politics and further strengthening the Illuminati's influence in the US.
This faction shares ties and overall goals with Hydra heads that took root in the Allies during and after WWII. They saw the US and Allies, broadly the West and allied nations, win both WWII and the Cold War. For them, freer societies are the best path to reaching strong and united humanity in the future. The primary points of contention between these broadly aligned factions is figuring out where to draw the line between enough freedom to ensure constant progress and control and protect humanity from itself. They're backing different ideologies in the West to determine what would work and what won't. Like many other Hydra-descended factions, those nominally allied with the US have gone at least in part native. They also pursue goals to strengthen the governments and countries they've been based in for generations.
2. The Hydra - the closest thing we have to the OTL movie Hydra faction. They're the epitome of Hard Men Making Hard Decisions While Hard.
Their end goal is to create a fascist world government led by Hydra. This faction would try to subvert Project Insight and use it to destroy their rivals and everyone who might oppose their takeover.
Splinter Hydra factions are vying for dominance of Russia after the SU fell apart, and its collapse crippled the primary Hydra faction that infiltrated and supported it. They're broadly allied with The Hydra out of spite of losing the Cold War and refusing to accept that heavy-handed oppression and control might not be the best way forward.
3. Smaller splinter factions: There are opposing Hydra factions vying for domination in China and India. At least one faction in each of those countries has gone native and wants future Indian or Chinese hegemony, respectively.
Smaller Hydra factions, or factions that originated as Hydra when the original group splintered could be found in many countries worldwide. While they broadly share the same end goal - united and strong humanity, they have as many ideas for getting there as there are factions. Some have entirely gone native, currently more interested in going after short-term national goals, and dealing with rivals, instead of pursuing Hydra's original objective. Various people and organizations in the Middle East and North Africa are poster children for this.
They're busy pursuing various conflicting secular and religious goals and interests. Such factions and associated groups could hardly be distinguished from governments, and parts of governments that aren't under Hydra's control or influence and pursue their own goals.