Fishing nets have mainly been contributed to an invention firstly by the Ancient Egyptians who wove nets and baskets from Willow branches. Simple boats were also made using Reeds.

Of course this is in debate to many however, as it is known that fishing is a far older, ancient practice that dates back at least to the Upper Paleolithic period which began about 40,000 years ago.

Like today the nets were commonly just thrown over the side of
the boat, or even just from someone standing next to a River.
Left still long enough, or slowly dragged towards the fish they
could capture several fish at a time which often served as a
meal for their families.

All kinds of fish could be caught depending on the size of the net, and the holes in it. These holes actually allow for smaller fish to swim through, while trapping the larger ones - ones more valuable.
Although it really depends what the fisher is trying to catch.

To the Egyptians the Nile perch, catfish and eels were among the predominant fish caught, and not only as part of a working day.
According to several documents found in papyrus drawings and illustrated tomb drawings fishing was also pursued as a
past time to those not already themselves cast into
slavery to work on the pyramids.

In comparison fishing scenes are rarely represented in ancient Greek culture, a reflection of the low social status of fishing.

A wine cup was discovered a number of years ago dating back to around circa 500 bc which displays a boy crouched on a rock with a fishing-rod in his right hand and a basket in his left.

In the water below, a rounded object of the same material with an opening on the top which has been identified as been a fish cage used for keeping live fish, or as a fish-trap itself.

The Wine Cup Image