How do negative and positive feedback loops occur and how are they different?

Think of them as a flow chart.

Stimulus (a change)
    |
Sensor (detects the change)
    |
Control (sends a signal about the change)
    |
Effector (response to the change) (connect a line to Stimulus)

Negative feedback is the action in which a certain variable is kept within a certain range of values.

One of the homeostatic mechanisms maintained by negative feedback is body temperature. In terms of the flow chart, if the Stimulus (body temperature) is too high, the Sensors (the hypothalamus in the brain or thermoreceptors in the skin) will detect it and send a nerve impulse to Control the Effectors (the muscle, skin and blood vessels in this case). The response from the Effectors will bring the Stimulus back down to within its normal limits. E.g. when you get too hot, you stretch out, sweat and your blood vessels enlarge (vasodilate).

This works the same way if the Stimulus is too low, except the Effectors will respond to increase the value again. For example, when you are cold, you shiver, curl up and your body hair stands on end.

So in short, negative feedback inhibits change from moving outside a normal range. So something controlled by negative feedback will continuously oscillate within that range.

Positive feedback follows the same process but the Effectors amplify the Stimulus.

For example, in the menstrual cycle, oestrogen is released by the ovaries. The oestrogen causes the brain to release GnRH and LH, which in turn stimulates the ovaries to produce more oestrogen. The cycle continues alongside the production of FSH until the LH and GnRH levels are high enough that ovulation occurs.