Welcome to Gaia! ::

Strangely enough, your cells are essentially colonies of independant organisms that became so specialised they couldn't live without one another.

The problem is if you go back far enough in time, you start encountering phenomena like plasmids and whatnot, and the trunk of the tree of life becomes a jumbled mess of interacting organic molecules. How these molecules specifically interacted and developed is a question for abiogenesis, and although we learn new stuff everyday, much of it is still a mystery. Here's to biologists.

And there's no question too basic so don't be afraid to ask. The only thing that irks us is people who wildly assert random falsehoods. Curiosity, on the otherhand, is always welcomed.

We represent all lifeforms evolutionarily as parts of a big "tree of life," where currently living things are like the tips of twigs, and we can trace back their ancestry like tracing back along those twigs to branches. A tree is a rather pertinent analogy because just as branches have lots of twigs coming off of them, so do past life forms often have multiple new life forms evolving from them.
So we are on one twig, and if you trace back, the twig we're on and the twig that, say, chimpanzees are on, join together, representing the period when we still had common ancestors. Then going further back brings us another joining, our common ancestors with, say, gorillas, and then monkeys, and then mammals in general, etc.

Since all of the branches must come off of a central trunk, we say that, metaphorically, the beings of this trunk must be the ancestors of all living organisms. As far as we can tell, they were probably very simple, single-celled organisms, things that we would barely consider alive by today's standards.

But then we get to the root system. In the roots of the tree of life are things that aren't alive yet, still below the threshold between life and non-life. Organic molecules, small interacting systems, eventually joining together at the base of the tree to form the first life. Unlike the branches, the roots are quite tangled into one another, and we haven't sorted them out very well because we can't see them very well at all, and they're all jumbled up: growing around each other and into each other and whatnot.
Abiogenesis tries to answer how this happened by looking at the period when the tree of life first broke the surface, and by trying to grow new trees of life now. Most of it is still highly speculative, though.

Well, to begin with, the idea of the first cell coming from other cells simply doesn't make any sense, so I think we can throw out that idea from the start. We know for certain that the first cells came from not other cells but from inorganic molecules, by the virtue of being the first cells.

Now, I think the cell theory that you're referring to is something akin to panspermia, which states that life on Earth actually originated in space and was transported here via asteroids hitting the Earth. Panspermia (in its basic form) makes no claims as to where those extraterrestrial cells came from, though, thus neatly sidestepping the actual question of the origins of life. It is plausible, but at the moment not very helpful.

I would have thought that prokaryotic cells would have evolved first, but yes, you are right.
On the other hand, while we're not completely in the dark, I'd say that the most important part of the cell, the information storage and replication system, is still somewhat up in the air. I wouldn't call anything without such a system a cell.
Anyway, if he is talking specifically about plant/animal cells, then his teacher ought to be shot for being vague. I was assuming that there was a more interesting miscommunication going on, but if it is just a distinction between plant/animal cells and cells in general, then I'll be very disappointed.
So, Karl, was your teacher actually talking about cells in general, or specifically the plant/animal cells, for strange but inevitably stupid reasons?

I can't wait to take psychics ^__^

If you meant physics here, then this is the funniest mispelling I've encountered in a long time.