How isomorphism makes groups alike? If two groups are isomorphic to each other they are same ( with different elements). But my question is when someone came up with the concept of isomorphism he or she must had checked some basic facts about those groups. But there are so many properties of groups; so on the basis of those few facts how could he tell that those groups were same?
 A: Anything that is a group property must be determined by the group's structure (which consists of a set and an operation).
So if two groups have indistinguishable sets and operations, they must be indistinguishable as groups (i.e. isomorphic as groups).
An isomorphism is a bijection (i.e. one-to-one and onto) function, so the underlying sets of two isomorphic groups are essentially the "same". 
For example: $f:\{1,2,3\} \to \{a,b,c\}$ where $f(1)=a$, $f(2)=b$, and $f(3)=c$. Even though the sets $\{1,2,3\}$ and $\{a,b,c\}$ are clearly different (one has numbers and the other letters) in a different way they are essentially the same as sets (they both are collections of 3 things). The "difference" between the sets is about being "letter" or "number" which aren't really "set" properties. So in some sense set theory is largely blind to the "differences" between my sets.
An isomorphism also preserves the operation: $\varphi(x\star y)=\varphi(x)*\varphi(y)$ if $\varphi$ is an isomorphism between two groups where the first group has an operation denoted $\star$ and the other group's operation is denoted by $*$.
This operation preservation guarantees that the two groups' Cayley tables match (if we had finite groups where we could write such a thing down). In words it says that if we operate then map, we will get the same answer as if we map then operate. 
This is enough to make sure that anything "group theoretic" said about one group will necessarily be true about the other group. This is because "group theoretic" statements just refer to the set (which we have made sure look the same) and the operation (which we have made sure operate the same).
I hope this helps!
A: Isomorphism is the natural choice of a notion of equivalence for groups.
A notion of equivalence is a way of deciding which objects we consider to be alike, and which ones we consider different. The finest such relation, among distinguishable objects is always the notion of equality ($=$), which distinguishes each object from every other object. But often times, there end up being many distinguishable objects of a given type that are, in terms of the structure of interest, alike in every way that is considered important.
For groups, the structure of importance is the binary operation on the set, and how the elements of that set interact relative to each other via said binary operation. It turns out that there are many different ways of constructing groups which result in a 'particular way' of elements relating to one another via a binary operation. Thus we are naturally lead to seek a notion of equivalence that captures exactly the things we consider important, and dismisses differences that we do not consider important. It turns out that the desired notion of equivalence is exact what you know as isomorphism of groups.
The above can actually be cast in more formal terms (in many ways in fact, but I will stick with a fairly simpler version). Consider a collection of objects (i.e. a set), $S$. A notion of equivalence for $S$ is none other than an equivalence relation for $S$, which we will call $\simeq$. Equivalence relations are binary relations on a set (that is, binary relations with domain=codomain), which satisfy the following properties:
Reflexivity: $\forall x\in S,(x,x)\in\simeq$ (every element relates to itself).
Symmetry: $\forall x,y\in S, (x,y)\in\simeq\Rightarrow(y,x)\in\simeq$ (if $x$ relates to $y$, then $y$ relates to $x$)
Transitivity: $\forall x,y,z\in S, (((x,y)\in\simeq)\land((y,z)\in\simeq))\Rightarrow((x,z)\in\simeq)$ (if $x$ relates to $y$ and $y$ relates to $z$, then $x$ relates to $z$)
These properties guarantee that an equivalence relation behaves as we would expect, it partitions $S$ into disjoint nonempty sets of objects. Each of those disjoint sets contains all of the objects that we consider to be mutually alike, every object in $S$ not in a given set of the partition is considered not alike to the elements in the given set of the partition. (note: this is something worth trying to prove on your own, try a few simple examples to get an idea how it will work in general. Moreover, comparing and contrasting different equivalence relations on the same set can be a rather interesting endeavour.)
It turns out that, given a set of groups, the relation, $\simeq$ defined by $G\simeq H$ if $G$ is isomorphic to $H$ (meaning there exists an isomorphism, $\phi:G\rightarrow H$. That is, a bijection, $\phi$, such that $\forall x,y\in G, \phi (x\star y)=\phi(x)\bullet\phi(y)$) is in fact an equivalence relation on $S$. Moreover, the proof of this does not use any information about $S$, hence the result holds for any set of groups (it's not a difficult proof either, it amounts to verifying the three properties above, for an arbitrary set of groups, $S$, which is done by showing that the required isomorphisms must exist).
One of the first things a mathematician who has defined a new object to study typically does is to try and come up with a natural notion of equivalence for that object, and much of the existing vocabulary of known mathematical objects have known corresponding natural notions of equivalence.
