Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps.
Regular Tessellations
A regular tessellation is a pattern made by repeating a regular polygon.
There are only 3 regular tessellations:
Triangles
3.3.3.3.3.3
Squares
4.4.4.4
Hexagons
6.6.6
Look at a Vertex ...
 

Semiregular Tessellations
A semiregular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same!
There are only 8 semiregular tessellations:
3.3.3.3.6
3.3.3.4.4
3.3.4.3.4
3.4.6.4
3.6.3.6
3.12.12
To name a tessellation, go around a vertex and write down how many sides each polygon has, in order ... like "3.12.12".
 
And always start at the polygon with the least number of sides, so "3.12.12", not "12.3.12" In order for a shape to tesselate, the interior angles must divide evenly from 360. Other Tessellations
There are also "demiregular" tessellations, but mathematicians disagree on what they actually are!
And some people allow curved shapes (not just polygons) so we can have tessellations like these:
Curvy Shapes Circles Eagles? 
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