From what I understand endorphins cause a release of dopamine via the mu-opioid receptor.
Dopamine is related with happiness due to it's effect on the reward system of the brain, in fact I believe they are the primary neurotransmitter involved with reward.
Serotonin can cause releases of dopamine but is also associated with happiness as it seems to be the neurotransmitter responsible for many emotions, and people with depression are shown to have a lack of serotonin.
there is some recent research (been told by a friend who read the original paper, so i can't find it now) that it looks like dopamine is mainly responsible for behaviour changes (especially repetitive behaviour/addiction, and it is released even when no euphoria occurs), while any happiness resulting from downstream endorphin activation.
serotonin released can also cause endorphin release.
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so basically all three neurotransmitter systems are interconnected any we have no idea if one is responsible for euphoria, if all three play a role or if happiness comes from some downstream activation circuit activated by all three.
what we do know is that all three are responsible for a wide range of activities throughout the brain (sometimes having more of one neurotransmitter acting on receptors in one part of the brain even has the opposing effect of having more of the same neurotransmitter act on the same subtype of receptors in a different area.) and (at least for serotonin and endorphins, dunno really about dopamine) in the rest of the body.
serotonin is an amino acid derivative and is mostly involved in mood, temperature regulation, perception, digestion, ....
dopamine is also an amino acid derivative and and involved in reward, expectation, mood, motor control, ...
endorphins are peptides respinsible for pain control, mood, digestion, ....
but have a look at wikipedia for a more complete description of what functions these compounds fulfill in our bodies.