Why doesn't Q posses Least upper bound property? What makes R different than Q so that R possesses least upper bound property. Q is also an ordered field.
Definition : 
An ordered field $(F, +, ·, ≤)$ is said to have the least upper bound property provided that any non-empty subset $A$ of $F$ that is bounded above has a least upper bound.
Lets take $(0,1)$ isn't 1 an upper bound? in $Q$
 A: Even though ${\mathbb Q}$ is "dense in itself" it becomes evident in mathematical praxis that there are important  numbers missing that we need, and are convinced "to be there", and which are not available in ${\mathbb Q}$. It's not just special values like $\sqrt{2}$, or $e$ and $\pi$, but in fact any infinite decimal which is not ultimately periodic, and there are many of these. The completion process invented by Dedekind  fills these fissures with a sort of "glue" so that in the end a perfect system results, which is not only "order complete", but allows of all the operations which we are so fond of from primary school arithmetic of fractions.
A: There are two separate questions here:

Why doesn't $\mathbb{Q}$ have the least upper bound property?

and

What about $\mathbb{R}$ makes it have the least upper bound property?


First question:
Note that for $\mathbb{Q}$ to have the LUB property, it's not enough for some sets to have least upper bounds (as you note, $(0,1)\cap\mathbb{Q}$ has a least upper bound) - every set needs to have a least upper bound. A good counterexample is given in the comments: $A=\{q\in\mathbb{Q}: q^2<2\}$ has no least upper bound, but is bounded above (e.g. $7$ is greater than every element of $A$). You can construct a number of other examples along similar lines.
This shows that $\mathbb{Q}$ doesn't have the LUB property; as to why it doesn't, one good reason is that it is just too small (even though it's infinite)! It's a good exercise to show that no countable ordered field can have the LUB property.

Second question:
It turns out that this question is slightly backwards. In fact, $\mathbb{R}$ is constructed specially so that it has the LUB! Basically, the idea is to take $\mathbb{Q}$ and "fill in" all the gaps: for every set $A\subseteq\mathbb{Q}$ which is bounded above, we want to add an upper bound of $A$; this gives us a bigger set of numbers, which (with a bit of thought) we can turn into an ordered field, and (with a bit more thought) show satisfies the LUB property.
This larger field - the completion of $\mathbb{Q}$ - is the real numbers! There are several ways to construct the real numbers - Cauchy sequences, Dedekind cuts, and others - but they all come down to starting with $\mathbb{Q}$ and "filling in" the gaps in some reasonable way. So the reason $\mathbb{R}$ has the LUB property, is just: we built it for precisely that purpose!
