Folklore & Mythology
Typically, mermaids are described as having the tail of a fish and the head and torso of a woman (or of a man, in the case of mermen). They are, of course, water elementals, and sometimes referred to as Undines. As spirits of water, they govern the West, though are themselves governed by the Moon. Depending on which part of the world you travel to, they are also called Merrows (Gaelic), Nixies (German), Rusalkas (Slavic), Amabies (Japanese), and Oceanids, Nereids, or Naiads (Greek). There are also selkies, seal-like creatures from Irish and Scottish folklore that can shed their skins and thereby transform into humans.
In folklore, mermaids are said to be vain, and are often seen lazing on rocks, combing their hair. Like sirens (who are often mistaken for mermaids but who are actually half-women, half-birds), mermaids may lead humans to their death by singing. In fact, mermaids are considered by sailors to be bad omens, though this wasn't always the case; to the ancient Greeks, Oceanids helped them to navigate the ocean, their father's domain.
As a mermaid enthusaist, I'd count myself fortunate to be visited by a mermaid in my dreams. Despite their vain and destructive reputation, they are one of the ultimate symbols of femininity, being that they're spirits of water and have close associations with Aphrodite, goddess of love and beauty (who herself was born of sea foam). However, many dream analysts would disagree with me. According to DreamStop, a mermaid is a bad omen, forewarning that you're about to be betrayed by someone around you or that you need to learn from past mistakes. A mermaid in your dreams (again, according to DreamStop) may also mean that a project you're working on is about to fail. Dream Moods, meanwhile, takes a similar, if more psychological, approach to mermaids:
To see a mermaid in your dream signifies the female aspect of yourself that is mysterious and secretive. It may also show a fear of sex. In particular, for a man to dream of a mermaid, it indicates that he is having fears of being drowned by the feminine aspect of his psyche. For a woman, it suggests doubts over her femininity. Alternatively, the mermaid may symbolize something very personal to you.
Working with Mermaids
Mermaids can be finicky, changing their moods with the tides. According to some, it's because their energies as elementals are pure and therefore they're reluctant to interact with humans directly unless their energies are pure as well. Either way, it may help to leave an offering. A mirror or ornate comb may appeal to their vain nature, or perhaps a pretty seashell or pearl to remind them of their home. You could also sing to them, or leave them something shiny or decorative, like jewelry or precious stones the color of the sea.
Info taken from Wikipedia, Elemental Awakening, What's-Your-Sign.com, The Love Witch, & Modern Tips for the Modern Witch
Blonde mermaid found at Webweaver Clipart
Ariel clipart and divider found at Ginevra's Disney Page