As web design is evolving and the requirements of a web visitor change at an alarming rate, the importance of focus on areas such as user experience and intuitive user interface design cannot be ignored. It is when a designed is restrictive, minimal and concise that the features of user experience are so apparent, not lost amongst a sea of visual flourishes, collages, shadows and gradients.
Typically I would associate a minimal style with the far east, japanese-pan-asian style of order, subtlety and discreet aesthetics, rather than maximal Vegas-style rah-rah in your face vibrant visuals. I would say that without the ability to hide under an assault of design touches and retouches, the minimal design by contrast can be that much more bold and impactive, as it is restricted to only a few crucial elements that must be carefully selected and precisely ordered to guarantee useability and success.
Clearly each design must reflect it's intention, a kid's site for instance must include a vibrant colour palette, big graphics and moving parts; clearly not a field to benefit from restriction and minimalism. Whereas typically 'portfolio' sites, highly corporate sites or those representing legal or accountancy firms can have a huge impact with a sharp and minimal style.
Some examples of great minimal designs are listed below.