The simplest way to describe minimalist shoes is to say shoes that mimic closest to running/walking barefoot. The idea behind why this would be beneficial is to say that "it's the way nature intended", however what has evolved is a discovery that it is not just a ideological fad, there is proof that it promotes a more natural walking and running form as users tend to organically adapt to landing on the midfoot and forefoot as oppose to landing hard on the heel without the large heel padding or incline from heel to toe, and all runners and experts agree (for or against minimalist) landing on the heel hard causes a lot of damage and is simply the wrong way to run. The second discovery is that it naturally promotes shorter strides with quicker steps that in turn actually propels the body more efficiently and reduces the shock absorption inflicted on the legs and body then say a large stride would cause.

Summed up into much simpler terms, it naturally trains the body to run and walk with better form.

Other benefits that proponents of minimalist would point out are more personal preference then "better for you" so to speak. Some simply prefer the extreme lightweight and added flexibility a minimalist shoe has compared to a traditional running shoe, noting that they move easier and more freely. Others note they get better control. Think of a rock climber who needs exact precision and grip in their footwear, with footwear that allows them to use their toes almost like fingers, not just one balled up "foot". Now compare the characteristics of a rock climbing shoe to a "minimalist" shoe.

Are there Downsides or Should I Toss Out All My Traditional Shoes and Replace Them?

As noted there are ample proven benefits, however that is not to say there is not cons and precautions to lookout for. Some experts would suggest minimalist shoes as an addition to a runner's arsenal of shoes, not an overhaul, at least to start. They do so noting that prolonged workouts and continued use of minimalists shoes can in fact cause damage if the user is not running properly due to the very minimal support. If running form is not corrected in a given individual and they are still landing hard on the heel, they are doing so now with less cushion. There are two popular schools of thought on this matter. One is to ease into it, to use minimalists shoes interchangingly with traditional shoes opting for minimalist more and more over time as form is corrected and worked on. Another is take the "right shoe for the right workout" approach where for running on hard surfaces such as road or concrete, stick to traditional shoes, where as dirt, trails, grass, sand, or cushioned tracks (indoor or outdoor) go with minimalist shoes. More and more experts are recommending having an array of shoes so this line of thought matches that notion (i.e. shoes for road running, trail running, gym workouts, muddy trails, rocky trails, hiking, cold weather, warm weather, the list goes on).

What Defines Minimalist Shoes?

Just because a shoes is light weight does not make it minimalist, and likewise just because a shoe is built for the trails does not make it not minimalist. Many brands focus more on the benefits and making a shoe that fits users needs best for a supported and healthy run without worrying whether or not their shoes are defined as minimalist, traditional, etc. but because of the benefits proven in minimalist shoes they may share many attributes of what can be defined as a "minimalist" shoe. Because of this there is actually a spectrum, some are shoes like Vibram Five Finger that are minimalist to their core, and even have started to be referred to as a subdivision of minimalist shoes, "Barefoot", where as some such as Topo Athletics or Inov-8 which are built on the principles of minimalist but share some characteristics of traditional shoes as well. What defines a shoe as minimalist really comes down to weight (it must be light weight), heel drop (zero drop is best, but usually anything 8mm drop or under can still be considered), flexibility (most minimalist shoes you can bend the toebox all the way to the heel and likewise can twist rotationally pretty effortlessly), and stack height (how thick the cushion of the sole is at the heel. Around 8mm is usually what minimalist look for but anything under 14mm can still be in consideration).