Guys, if I may be permitted to add to the OT.
The grain pattern seen on veneers is due to the growth of the girth of the tree in different seasons over the years. In some trees this can clearly be seen as 'Annual Rings'
Decorativve veneer, as the name suggests, is a slice of a tree having distinctly visible grain pattern pasted on to the furniture purely for aesthetics. To get the pattern, log has to be sliced the way it is shown in the video linked by GeorgeO.
The second one linked by Thad is used for making plywood purely for functional reasons. Since wood has tendency to expand and contract, layers of wood extracted this way are bonded with alternate layers perpendicular to each other by using adhesive resin and pressed together in a heavy press to form sheets of plywood. Since the layers are perpendicular to each other, they expansion and contraction of each other gets cancelled.
As regards the thickness of the veneer, as George and someone else said, it is less than 1mm. However, in India, commercially available veneers are 4mm thick where, the layer of veneer is factory glued onto a 3mm thick commercial ply. For the record, I have samples of both.
Hope this clears the confusion.