A 2D Ocean around a Circular Planet
While the famous Flatland portrayed a world where gravity was effectively irrelevant, it is very unlikely for life to form on an endless, uniform plane for the same reason that life doesn't form in outer space. For complex chemical reactions to take place, you need a physical force that pulls distributed particles close together, trapping them and forcing them to interact. In other words, you will need a 2D analogue of gravity to create 2D analogues of planets. Some of these planets may have 2D analogues of oceans, with liquids that particles can float around in. In such conditions, life - or something like it - may evolve.
It will be very difficult for 2D life to operate on land. Since the surface of a 2D circular planet is a 1D line, the only way for land animals to get around their neighbors will be to climb, jump, or fly over them. This is inconvenient at best, and makes complex social interactions nearly impossible. So full-time land-dwellers will probably be very rare, and if sapience ever evolves it will probably be in the ocean.
Jigsaw critters with multiple brains
At first glance, it might seem as though these organisms would be unable to have more than a single opening, with branching tubes to distribute nutrients throughout the body. After all, anything else will cut the organism in half. For most 2D organisms, this might be the case - but there is another option.
Picture a creature that is basically a jigsaw puzzle - multiple parts, but those parts are shaped in such a way that they "snap" together and are unable to be pulled apart without deforming them. If the spaces between the pieces are relatively large, but smaller than the curves and angles holding them together, it is possible for a 2D creature to have an anus and a circulatory system.
What is particularly interesting about such a body shape is that each "segment" of the body will probably have a separate nervous system. This may mean that each creature has two (or more) separate brains that work together, or it may be possible for it to have nerves that can actually split apart and re-join.
Jellyfish and Flip-Fish
On our world, most advanced life forms have heads, mouths, and anuses. They have a front side that is distinct from their back side, and a top side that is distinct from their bottom side, while their left and right sides are usually similar. This will not be the case on a 2D world.
On a 2D ocean world there are two "equivalent" directions (left and right) and two "non-equivalent" directions (up and down). On our world, most mobile animals developed a distinctive "head" and "tail" end - the head end in the direction of motion (cephalization) - as well as a dorsal and ventral side, with the left and right sides being symmetrical.
But on a 2D world, most animals will have one symmetrical axis and one non-symmetrical axis. While one might think it is advantageous to have a front, back, top and bottom, it is impossible for a 2D "fish" to turn around without flipping over. This means that such a creature would have to always be on the "correct" side of whatever it is pursuing - rather inconvenient.
This means there will likely be two main types of cephalization on this 2D world. The most obvious will be "jellyfish-like" - they will have a "front" on either their top or bottom, and basically "hover" around, moving left or right but rarely flipping over. They will have a vertical axis of symmetry, their right and left being basically identical. They will probably be relatively slow, controlling their motion by adjusting their buoyancy, hunting prey by either ascending toward it or descending on top of it.
The second type has no simple analogue on our world, but I will call them "flip-fish". These also have a single axis of symmetry, but it will be their horizontal axis instead. These have a distinctive "front" and "back" side, but their "top" and "bottom" will be symmetrical. They will swim quickly horizontally, and will "flip over" in order to change direction. These animals will probably be quicker than the "jellyfish" on average.