This is a complex topic, but here is a go at it.
Suffering exists because of our desires (cravings), aversions and ignorance. Now it should be explained that the Pali word the Buddha used is Dukkha. As in all translations some words can't be translated one-to-one from one language to another. Dukka in Pali suggests a basic satisfactoriness with life, due to the fact that our lives are impermanent and constantly changing. So it means unease, disquiet, stress, not really suffering, but suffering sort of captures the overall meaning the Buddha was trying to describe in English.
The Buddha explained that our suffering doesn't come from without, but from within. That our desire to be taller, shorter, richer, cooler, drive a better car, live in a different place, etc (Desires). drives us to be unhappy. Also the fact that we think life should be pain free, without fear and want (aversions) makes us unhappy. Finally not realizing how the world actually is and how it really works (ignorance) makes us unhappy. All these things are controlled within our mind.
Think of this example, two people are waiting for a late bus, one is fuming, and the other is whistling a happy tune. Why? The bus is late for both, but one is suffering, the other is not. Is the bus the problem or the person who is mad over something they can't control? Is getting mad going to make the bus come any faster?
Finally, there is a mantra I use to help remind me of how our minds control how we react. It is:
All experience is preceded by mind, Led by mind, Made by mind.
Speak or act with a corrupted mind, and suffering follows as the wagon wheel follows the hoof of the ox.
All experience is preceded by mind, Led by mind, Made by mind.
Speak or act with a peaceful mind, and happiness follows like a never-departing shadow.