(I'm no mathematician so bear with me. Tessellations are a complex subject so I'll try to keep it short.)
A tessellation is a repetition of a "tile" (one or more geometric shapes with no gaps or overlaps) in a plane to form a pattern. There are several types of tessellation patterns. For example a honeycomb is a regular tessellation (all sides and angles are equivalent):
The mosaics in the Alhambra palace are non-regular tessellations (there are no restrictions regarding the shapes used or their arrangement around vertices):
Intricate shapes have many elements arranged in a complex manner; in this context it just means that it has complex details. It's a single image with complex shapes. It will only be a tessellation if
So your tessellation can be composed of tiles with intricate shapes.
Look at the example below. It's both composed of intricate shapes (each bird tile) and it's a tessellation semi-regular (there are two distinct bird tile shapes). Bonus: the bricks on the wall can also be considered another tessellation pattern:
Wikipedia also tells us:
A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged.
... A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern.
There's more info here.