The basic premise is there are two types of mindsets, a fixed mindset and a growth mindset. Those with the fixed mindset are those that believe that intelligence, ability, etc.. is basically fixed. You have some certain amount and that's that. The growth mindset believes the general range of these things may be strongly influenced ('fixed') by things like genetics, life circumstances, etc... but improvement within this range is definitely possible and the key to that is hard work and honest assessment of where you are relative to where you want to be.
This relates to the achievement vs. hard work thing because she claims children who are praised for 'being smart' or 'being good at X' tend to gravitate towards the fixed mindset (i.e. 'I get praised because I am smart, I am smart because I can do X well/X comes easily to me, if something doesn't come easily to me it must mean I am not smart/talented'). This causes them to not put in effort when the going gets tough and in fact to avoid challenges because if they fail they view it as a judgement on their core self/competency (not simply an indication of an area for growth).
The growth mindset folks (children praised for doing well because they worked hard at it as opposed to some natural talent or 'smarts') tend to seek challenges as they view them as the engine of growth/improvement.
Using these frameworks as a lens on which to view human behavior can be interesting. I have definitely seen both mindsets in action (in myself and others). I definitely , consciously, try to stay in a growth mindset now, but I think our culture heavily pushes a fixed mindset where someone either has 'it' or they don't, they are smart or they are not, they are talented or they are not. We prefer the 'instant success due to massive talent/smarts' story over the 'worked their ass off for years to build amazing talent and then succeeded due to that hard work'.
The book Talent Is Overrated also touches on this and points out most people that we generally consider 'naturals' at things, if you interview them/study them/look at their past, all have something in common, a tremendous amount of effort in learning/training, above and beyond what most people put in. This also veers towards the 10,000 hour theory of Anders Ericsson
EDIT: Fixed a bunch typos/misspelling I saw in re-reading. Originally typed in IE with no spell check. Area for improvement: spelling.