Groups are used in studies of symmetry - in Math and in the sciences, especially in Physics and Chemistry.

A group is basically a set $G$ of objects that can be "combined", so that two objects $x, y$ in $G$ produce a third object $x \ast y$ in G. Loosely, we refer to this combination or operation as a 'multiplication' (or it could be an 'addition'). This operation has to have three basic rules:

1. The operation must associative, i.e. $(x\ast y)\ast z = x\ast(y\ast z)$ for all objects $x, y, z$ in $G$.

2. $G$ contains a special object $e$ such that $e\ast x = x = x\ast e$ for all objects $x$ in $G$.

3. Each object $x$ in $G$ has an associated object $y$ such that $x\ast y = e = y\ast x$.

Condition 2 says that the object $e$ has no effect on any other object - it is called the "identity" object. It behaves much like the real number 0 in relation to the addition + operation since $x + 0 = x = 0 + x$ for all real numbers. (Here, in this example, $\ast$ is addition + and e is 0.) As a second example, $e$ could also be the real number 1 if $ast$ stood for multiplication (in which case we take $G$ to be all real numbers except 0).

Condition 3 says that each object has an 'inverse' object. Or, that each object could be 'reversed'. It turns out that you can show that the $y$ in condition 3 is unique for each $x$ and is instead denote by $y = x^{-1}$.

The commutative property -- namely that $x\ast y = y\ast x$ -- is not assumed, so almost all of the groups in the periodic table do not have this property. (Groups that do have this property are called Abelian or commutative groups.) The Abelian simple groups are the 'cyclic' ones that appear in the right most column of the table. (Notice that their number of objects is a prime number $2, 3, 5, 7, 11, \dots$ etc.)

The periodic table lists all of the finite simple groups. So they are groups as we just described. And they are finite in that each group $G$ has finitely many elements. (There are many infinite groups used in physics but these aren't part of the table.)

But now what are 'simple' groups? Basically, they are ones that cannot be 'made up' of yet smaller groups or other groups. (More technically, a group $G$ is said to be simple when there isn't a nontrivial normal subgroup $H$ inside of $G$ -- i.e., $H$ is a subset of $G$ and is also a group under the same $\ast$ operation of $G$, and further $xHx^{-1}$ is contained in $H$ for any object $x \in G$.) So a simple group is like a basic object that cannot be "broken down" any further, like an 'atom', or a prime number.

One of the deepest results in the theory of groups that helped in this classification is the Feit-Thompson Theorem which says: each group with an odd number of objects is solvable. (The proof was written in the 1960s and is over 200 pages - published in Pacific Journal of Mathematics.)