For background on A_{n} and S_{n}, there are many good resources on the internet, here is one:
Lecture 2.3 Symmetric and Alternating Groups

In this document, permutations are represented as dots with arrows. The arrows show how the dots rearrange. The tables show the product of any two permutations by column/row-lookup. The permutation of the column is applied first. Apologies if that convention is confusing, I had to pick one, and that one was intuitive to me.

A_{2} is the trivial group

S_{2} is isomorphic to Z_{2}

Here is where things start to get interesting. A_{3} can be thought of as the orientation-preserving symmetries of a triangle, or as Z_{3}.

S_{3} can be thought of as the full group of symmetries of a triangle (a dihedral group). This is our first non-commutative example. In this table, it matters whether you look up column-first or row-first. To my mind, the order of composition is column-then-row.

This group can be thought of as orientation-preserving symmetries of a tetrahedron. To see this, think about how a symmetry of the tetrahedron permutes the corners. For example, consider an entry in this table with one fixed point and the other three points in a cycle. Visualize that as a tetrahedron rotating 120-degrees.

This group can be thought of as orientation-preserving symmetries of a cube. To see this, imagine antipodal corners of the cube connceted by a line. Associate the four lines with the four points in each permutation.

This group can be thought of as orientation-preserving symmetries of a dodecrahedron or an icosahedron. A_{5} is also an example of a non-commutative finite simple group. For more background on finite simple groups, see this document: Classification of Finite Simple Groups

And finally, here is S_{5}. This group has 120 elements, so the table has 14400 entries.