what is meant by minimum element ? what's the difference between minimum and minimal element? We say that $x ∈ S$ is the minimum element of $S$ (with respect to the generalized inequality $≼_K$) if, for every $y ∈ S$, we have $x ≼_K y$.  
We say that $x ∈ S$ is a minimal element of $S$ (with respect to the generalized inequality $≼_K$) if, for every $y ∈ S, y ≼K x$ we have $y = x$.
We can describe minimum and minimal elements using simple set notation. A point $x ∈ S$ is the minimum element of S if and only if
$S ⊆ x + K$.
Here $x + K$ denotes all the points that are comparable to $x$ and greater than or
equal to $x$ (according to $≼_K$ ). A point $x ∈ S$ is a minimal element if and only if $(x − K) ∩ S = {x}$.
 A: Your definitions are incorrect.

Something is minimal if nothing is smaller. Formally we write this
  as "$y$ is minimal if $\forall x, y\leq_K x$."
Something is the minimum if it is the unique minimal element.
  Formally we write this as "$y$ is the minimum if $\forall x(x\leq_K
 y\Rightarrow x=y)$. Alternatively, we can write "$y$ is minimum if $\forall x(x=y\lor y<_K x)$," where $a<_Kb\iff a\leq_Kb\land a\neq_Kb$.  Notice that the equality in both cases is actual equality, not equality in the partial ordering induced by $K$

Let's look at an example. Consider $\mathbb{C}$ equipped with the partial order induced by $|\cdot|$. The minimal points of $\{z:3\leq |z|\leq 5\}$ are the set $\{z:|z|=3\}$, but the function $f(z)=|z|$ has a minimum value of $3$ on that set. When talking about the output of the function, there is a unique smallest output, and so it's the minimum. When talking about the input there are multiple points that give rise to the smallest output, so those are minimal points.
A minimum point is always minimal, but the converse statement is not true in general. However, it is true for total orders.
A: You have almost correctly quoted the formal definitions (see @StellaBiderman 's answer), so I assume you're asking for the idea, expressed more in words.
"Minimum" means "smallest". In the usual ordering of the natural numbers $\{1, 2, \ldots \}$, the number $1$ is the minimum.
"Minimal" means "nothing is smaller". If you order the natural numbers starting with $2$ by divisibility, so $a "\le" b$ when $a$ divides $b$ then the minimal numbers are the prime numbers. $7$ isn't minimum - it's not smaller than everything else - but it is minumal - nothing is smaller than it.
