This is a drastic oversimplification, so take this as a generalization rather than solid fact. You can click on the image icon for each domain below in order to read the storyline history of that domain.
Sylvan - This domain can focus on a rush strategy, playing cheap fast creatures and attacking your opponents base before they have a chance to really get going, while placing cheap buildings to claim flux wells. Sylvan has a few 3 domain bombs, but is slow to get high flux and high domain. As you might expect, Sylvan also lends itself to forest-centric decks, which is a theme some players try, especially with Loyal Wilds.
Mountain Folk - Dwarves tend to be inventive, hard working, and strong. These aspects are the basis of Mountain Folk. There are many styles of Mountain Folk, from a rush-type strategy to building strong 3 domain armies with the domain acceleration of Hadarack's Throne. MF also lends itself to control/counter decks.
Dark Legion - This domain's strength is with flux acceleration and domain acceleration, allowing one to cast amazingly strong 3 domain creatures and enchantments. It has a good dose of control as well. A theme deck with this domain is Vampires.
Elemental - The Elemental domain can be a strong domain when played solo, especially if built around cards like Meld, however most players use it as a supporting domain in two domain decks. It has control, fast creatures, and lots of TRICKS, which can be quite fun!
Chaos - The name says it all. This domain is capable of producing chaotic hell against enemies, though it can do the same to you as well. Valid strategies with this domain have been well rush, creature rush, control, and 3 domain bombs.
Celestial Imperium - With the use of Trade Routes, this domain is the fastest to get to 3 domain bombs, though the domain acceleration also helps your opponent. Great 3d creatures and buildings help to justify this risk. On the complete opposite end of the spectrum, this domain can also have 1 domain creature rush with a well placed Rune of the Swift. Psion-themed decks can be quite strong.As you can see from the above, this game is full of strategies, and each domain can usually lend itself to each strategy you might want to try.
Welcome to the game, have fun, and give each domain and strategy a shot until you find one that fits your personality and gets you excited to play.