In an easy-to-understand way, imagine that you have a tap that sprays water out. If the system is at a low pressure you get a sort of gentle trickling effect, but if it is at a high pressure then you'll get a powerful stream of water coming out of the tap. Your heart is a pump and all your blood vessels are the system. Within that system there is a resting pressure. Hypertension is where that resting pressure is too high. Your heart has to work a lot harder to pump blood out against a system that's so high pressured and it adjusts its physiology to some extent to do so.
As for why resting pressure becomes too high, the most common reason is to do with the system itself. Water pushed through a tiny hole is going to come out at a higher pressure than water pushed through a wide bit of hose, and it's the same with blood. If blood vessels become narrower then pressure rises. Blood vessels also have some elasticity to them which reduces pressure because they can sag out slightly as the pressure gets higher. If they lose this, pressure gets higher.
Over our lifetimes, damage to our blood vessels happens to everyone, and to some people more than others because certain factors increase the amount of damage - smoking, diabetes, dietary factors such as cholestrol, alcohol consumption - and as they become damaged and the elasticity falls and the vessels get narrower, blood pressure goes up.
It's a complicated system, there are a lot of ways in which pressure goes up and down. For instance in old age, despite thinking that actually damage is cumulative and so it should be the HIGHEST blood pressure, we actually drop our blood pressure. So you can see it's complicated. Issues with the pump, with blockages, with reflexes that help us maintain sufficient blood pressure, with the areas perfused by the blood such as the liver, kidneys and so on... many many things! But in terms of issues with the classic sort of hypertension you see in middle age, that's the most common scenario.
Hypertension is dangerous for various reasons, largely because it increases your risk of strokes, it seems to be a negative factor on survival in multiple conditions and has profound effects on your physiology as it adapts to the higher pressure. For instance you can have hypertrophy of the muscle of the wall of the heart, which seems quite logical except that it reduces the amount of space inside the chambers of the heart for blood and means that the resting pressure actually increases as well. This is also complicated :P