I think it's important to understand what mass actually is. It's no more or less than resistance to motion. Newton's second law:
Force = mass * acceleration.
or, rearranged,
acceleration = Force / mass
i.e. for the same applied force, the object with more mass will accelerate less.
Now Newton's second law is not correct at high speeds or for subatomic particles; then we need relativity and/or quantum mechanics. But the idea of mass as resistance to motion remains the same.
You're very familiar with resistance to motion in the context of fluids. Try pushing a ball through water rather than air. Here the viscosity of the water resists the motion of the ball.
The Higgs mechanism for mass is a little like the fluid example above. The entirety of space is permeated with these Higgs bosons that appear and disappear randomly, but on average some are present. Other particles interact with the Higgs bosons and it's this interaction which, just like the viscosity of the fluid above, tends to resist the motion of the particles. Moreover, the Higgs bosons interact with themselves.
The interaction is actually rather complicated, not nearly as simple as a viscosity term. CERN has a better analogy:
"Imagine you're at a Hollywood party. The crowd is rather thick, and evenly distributed around the room, chatting. When the big star arrives, the people nearest the door gather around her. As she moves through the party, she attracts the people closest to her, and those she moves away from return to their other conversations. By gathering a fawning cluster of people around her, she's gained momentum, an indication of mass. She's harder to slow down than she would be without the crowd. Once she's stopped, it's harder to get her going again."
And Jorge Cham of PhD comics made this cute movie:
http://vimeo.com/41038445Hope all this helps a little.