Do I need to play the original Mass Effect game to understand the sequels?

I was in your shoes this past January. I decided to buy the Mass Effect Trilogy and just go for it (I'm currently half-way through ME3 after finishing ME1 and ME2).

I highly recommend that you play all games in order (ME1, ME2 and ME3) since your choices will carry on between games if you import the save files. Your choices in Mass Effect 1 will impact Mass Effect 2 and 3. Your choices in Mass Effect 2 will impact Mass Effect 3 and I'm guessing that these will somehow impact the upcoming Mass Effect: Andromeda.

This makes for a better overall storytelling experience as you can really feel your actions changing the universe of Mass Effect.

You should also consider buying the story DLC for all 3 games for a more complete experience. As Aventinus mentions, some of the story DLC fills important gaps in the story that can feel a bit lacking without them.

Keep in mind that you can't purchase Mass Effect 3 from Steam though. However (as Bowdzone mentioned) you can import your save files regardless of the platform you bought the games from (e.g. you can import your save files from ME2 to ME3 even if you bought ME2 on Steam and ME3 on Origin).

Both ME2 and ME3 have an optional comic-style introductory sequence (resp. Genesis and Genesis 2) that allows the player to (just barely) catch up with the previous scenario (ME1 on PS3, ME1+2 on Wii U) as well as make a limited selection of key choices from the previous games. Genesis 1 and 2 are available for all platforms but as DLC or directly on-disc depending on platform.

Yet, to enjoy the full breadth and "universality" of the game, an interested player should play all games by chaining them together. This is especially true for ME3 which stands as the dramatic conclusion to the Shepard epic, where many quests from the previous games are consequential to various major and minor events that play out during ME3. People complained about ME3's peculiar ending, without realising that the whole game is effectively the ending of the full story arc. Also, bonds you create with your party are not as impactful when you didn't travel the whole galaxy and back at their side, from saving each other's neck to having a party to growing a friendship or love affair.

That said, from a gameplay perspective, the ME1 engine, inventory, and controls are significantly different from and more dated than ME2 and ME3, and somehow much closer to KOTOR despite the obvious graphics quality bump. An unwary, casual player may be turned away by this challenge, especially if coming from the action games side of things and therefore may be best served by playing Genesis+ME2+ME3.

To answer your question directly, no, you do not NEED to play the prior games to enjoy ME3, it has a good story that can stand on its own.

With that being said however, for all the reasons stated in the other answers, you absolutely should play them in order, with all DLC. It will make for a much more enjoyable experience, and explain some of the subtle story arcs (Like Jokers story) and in-jokes (Make sure you take Shepherd to the club in each game. You're welcome)

Also, when you do play through ME1, please do yourself a favor and wait until AFTER the Citadel to decide whether you like the game or not... The Citadel portion can be very long and boring (Those elevators are killer!) and is not indicative of the how the majority of the game plays, it helps to set up story arcs and get you familiar with the RPG mechanics