All these links are accessible at a non-mathematical level, and they are by recognized scientists (with the exception of the first link).
(1) To start, see the "Simple English Wikipedia", which explains what the Higgs effect is, and the reason for the Higgs effect: http://simple.wikipedia.org/wiki/Higgs_field.
(2) The difference between the Higgs boson and the Higgs field is in this short answer from Fermilab: http://www.fnal.gov/pub/science/inquiring/questions/higgs_boson.html.
(3) Here is a complete explanation of the Higgs field, the particle, and how the field works, written by a theoretical physicist: http://profmattstrassler.com/articles-and-posts/particle-physics-basics/how-the-higgs-field-works-with-math/. Here is how it impacts the standard model: http://profmattstrassler.com/articles-and-posts/the-higgs-particle/the-standard-model-higgs/.
(4) From the Quantum Diaries website, here is an explanation of the Higgs boson with Feynman diagrams: http://www.quantumdiaries.org/2011/03/25/an-idiosyncratic-introduction-to-the-higgs/. This is a more complete series of Feynman diagrams, which requires a good deal of attention: http://www.quantumdiaries.org/2011/05/05/a-diagrammatic-hint-of-masses-from-the-higgs/. The Higgs boson in the vacuum: http://www.quantumdiaries.org/2011/06/03/higgs-and-the-vacuum-viva-la-vev/
(5) Instructions on how to explain the Higgs Mechanism to others, from a scientist's blog: http://www.preposterousuniverse.com/blog/2012/12/07/how-to-explain-the-higgs-mechanism/
(6) Here is another view, which also tries to explain the Higgs boson with a Feynman diagram. It's interesting because the Lagrangian for the Standard Model is reproduced in full. You can see all the math, and the Higgs part is circled in red, and explained: http://www.quantumdiaries.org/2012/09/13/higgs-problems/. This is a good attempt to place the Higgs in the Standard Model, while keeping it accessible to non-mathematicians yet giving you a taste of the math.