I know that life on other planets doesn't have to resemble anything on Earth, or even be compatible with it, as life could have developed in completely different ways, but would they be made up of cells? Would animal/plant cross things be possible? What would these things breathe? Other notable qualities?
closed as too broad by Ghanima, Scott Downey, bowlturner, DaaaahWhoosh, James♦ Mar 3 '15 at 14:48
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Energy absorption: Earth life is based on a very impractical concept of continually absorbing oxygen to support burning reactions of metabolism. Why not photovoltaic? Why not nuclear? Digestion and photosynthesis aren't the only option.
The weird 'second step' where first there's a duplicating cell, then there's a duplicating clump of cells (organism) is absolutely unnecessary. As cells develop various specialized organelle, like mitochondria or flagella, who says they couldn't develop more complex ones - far more complex ones? Or symbionts that depend on each other to live but don't bind physically: a set of multiple distinct organisms that can't live alone (for long) but only together form a set capable of sapience (say, 'brain part' being immobile and fed by 'forager' part which accepts instructions, say, by sound modulation).
Post-organic; artificial species. We may imagine and develop arbitrary limits on what could evolve naturally, in random conditions like the aminoacid soup ocean of early Earth, but you must always allow for engineered races, built - self-repairing, self-replicating robots, experimental 'alternate biologies' (that slipped out of control), powerful non-sapient computers that still run sapient programs, 'alternate robotics' (e.g. based on quantum computing, on biomechanical hybrids etc), and countless others that couldn't 'just happen' but could have been built, and then somehow keep existing (and evolving, far from original!) apart from their creators.
Beyond limits of our science. We can only picture biologies within frame of what we know - e.g. about down to quark level. We don't even imagine many concepts which are yet to be discovered - and may be basis for forms of life.
Sure animal/plant like organisms would be entirely possible, because some organisms actually exist here on earth that already blur the lines between the two.
There is the Mesodinium chamaeleon which is classified (by some at least) as part animal part plant, but it is a simple celled organism, which probably does not fit the concept of what you have in mind. (Also check out "Euglena")
There is the Green Sea Slug which also exhibits both animal and plant qualities. Obviously a slug is an animal, but unlike other slugs it produces food by a form of photosynthesis using chlorophyll just like a plant. So that answers part of your question about what they might eat.
I believe I remember reading about some other organisms that have been classified as both, but are perhaps extinct now.
You might also want to consider viruses. They are usually thought of as not being alive, but yet they "reproduce" or rather use the mechanics of a living cell to cause reproduction. They have DNA or RNA, even outer layers of proteins and fats, but still don't exhibit life as we usually define it. They don't really eat, just invade other cells and then produce new viruses. Yet one could easily imagine a more advanced virus like organism that might act much the same way, and yet have intelligence, and be able to move about on there own.
They need not have cells , actually you have to redefine life, your definitions based on human characters may lead you to feel there is no life elsewhere , and you will boast yourself as the only lifeform in universe