Hormones interact through both responding to the body's needs and through regulatory feedback. Some hormones are generated directly by the cells that respond to a condition in the body. For example, insulin is produced directly in response to heightened blood sugar by pancreatic beta cells.
Often, though, hormones are secreted in a more complicated fashion. The hypothalamus will put out releasing hormones to tell the pituitary to respond to a given condition. For example, if the body is stressed, the hypothalamus will secrete CRH, corticotropin releasing hormone. In one case, the hypothalamus is able to regulate itself, where growth hormone releasing hormone (GHRH) inhibits its own release.
The pituitary is directly stimulated by the hypothalamus, in general, to release tropic hormones. These hormones are then sent throughout the body to either perform a direct stimulus, like with Growth Hormone (GH), or to stimulate another organ to produce an end hormone, like when CRH stimulates adrenocorticotropic hormone (ACTH) to stimulate the adrenal glands to produce hormones. Pituitary hormones are also used to give feedback to the hypothalamus. This "short-loop" feedback tells the hypothalamus that the hormone signal has been received by the pituitary, and the hypothalamic release of releasing hormones in many cases is inhibited.
End hormones are the final effector hormones stimulated by pituitary hormones. These hormones act directly on the target tissue to produce the desired effects. For example, thyroxine is mobilized by thyroid stimulating hormone (TSH) to move throughout the body to increase cellular metabolism. End hormones also act to downregulate secretion by the hypothalamus and pituitary (sometimes both, sometimes only one or the other) through "long-loop" feedback. For example, thyroxine downregulates both the release of TRH from the hypothalamus and TSH from the pituitary.
There are a few notable exceptions to this system. The first is where an end hormone stimulates the hypothalamus to secrete a factor. Insulin-like growth factor 1 (IGF-1), one of the end hormones for GH, acts to stimulate somatostatin (SST) release from the hypothalamus. SST then inhibits GH release from the pituitary. The second exception is where the hypothalamus normally sends signals to inhibit secretion of prolactin (PRL) through dopamine (DA) secretion. Finally, there are some positive feedback loops for certain hormones. For example, oxytocin released from the pituitary causes uterine contraction, which causes more oxytocin release, which causes more uterine contraction, which will eventually cause a baby to be born.
There is a lot more information on this subject. In fact, hormone interactions will fill textbooks! However, this gives some of the basic ideas behind their interactions.