As Aslan is C.S. Lewis' version of Christ, so Tash is his version of Lucifer or Satan. Lewis did not specify about exactly what Tash is or where he came from (or how he came, for that matter) and so this fan-based article is an attempt to explain.
Long ago, there were no worlds but Aslan's country. I am not sure when this was, but it was a very long time ago. However, the Emperor-Over-The-Sea (as he is called in Narnia) had been planning for an even longer time for other worlds to be made. He had a son (very crudely speaking), Aslan. Now the Emperor had created other beings who were less in power than Aslan but nonetheless powerful. These beings were to help bring order to the countries the Emperor was going to make.
The very first place the Emperor created was a great forest, which was to be the Wood Between the Worlds encountered by Digory, Polly and the other characters from The Magician's Nephew. This place was not technically a world, as our world or Narnia is, but it was to be rather a link between all the worlds. This infinite forest ringed around the great Mountain of Aslan, and sat on top of the other worlds (hence when a person stepped into one of its pools with a special ring on, he would "fall into" one of the worlds).
Now once the Emperor had finished this "in-between", he set Aslan to start making the worlds. For those who have not read The Magician's Nephew (which is about the beginning of Narnia), I will give a brief explanation. Each of the worlds started out as just a black emptiness. Aslan then started singing, and in a nutshell this song created everything that comprised that world. Every world was at its start quite perfect, with not a single flaw. In fact, it would have continued this way, but unfortunately something terrible was about to happen.
Now each of these beings went to and fro from Aslan's country into the newly created worlds, to help bring order to the creatures that had been formed to populate them. They took different forms; great reptilian beasts (like the kind Jill, Eustace and Puddleglum saw in the Emerald Witch's kingdom), giants (like Father Time), and a great number of other beautiful creatures. There was one, who was the most powerful of the beings and who had been appointed to help Aslan in the Great Songs which started the worlds. Now, as more worlds were created, this being (who was not specifically named) grew envious of what had been brought into being. He greatly desired these worlds for himself; in fact, as time went on he came to desire to govern each of them, and even Aslan's Country itself!
This envy gradually began to rot the being from within, and he began to lose his glory and a great blackness and horror overtook him. Soon, he became the epitomy of evil itself, and he was able to corrupt other of the lower beings (one of which married a mortal creature from Charn, and thus began the semi-immortal race that ended in the Empress Jadis). Now the Emperor knew about this already, and so he set about the creation of a different place, one which was totally different from his country or any of the worlds which were being created. It was a dark place of misery, terror and torment. And the Dark One was henceforth known as Tash.
Now the Emperor punished Tash by binding him to this terrible dark place. However, it was not a strong boundary, as Tash was allowed to briefly enter the other worlds, to test the hardiness of the creatures there. Unfortunately, the creatures of all the worlds, hardy as they may have been, were not a match for Tash's influence and destructive power. He was able to convince many that Aslan was tricking them and had awful plans for them, or that Aslan simply didn't exist and their worlds had no purpose and were there by chance instead. One of the kingdoms he corrupted was that of Charn. Now the people here were much bigger than those of our world, which was one of the first to be created. Aslan had granted these people the gift of magic, but when Tash visited them and saw such potential, he corrupted them and their magic.
Once the Emperor had made this country, he created links to it from all worlds, though there was no link to it from Aslan's Country (Aslan created a temporary link at the end of The Last Battle with his shadow). Therefore, those who had conformed to Tash's ways were then assigned to this horrible place, to spend eternity. However, he did not wish for them to have to go here, so he created a way for them to come out of the Dark One's corruption, if they would. But that is not discussed in this article. Here, I am simply describing the place.
Tash's Country is the polar opposite of Aslan's Country. Now I have not been to this place and have no wish to see it, but Aslan had been graceful enough to tell some of the worlds something of it, so I am writing down what I know.
This country is essentially the same as the version of Hell that is portrayed in The Great Divorce: Dark, gloomy, meaningless and agonizingly dull. There is no cheer or happiness here; just utter loneliness and regret. Every punishment in this place is self-imposed; the bad qualities of the condemned in life are simply made worse here, and thus prevent that creature from experiencing any kind of happiness.
This country was assigned to the great traitors and other followers of Tash, and Tash was able to send these creatures to his country with a simple peck, as he did to Shift the ape in The Last Battle. During the early years of the world of Narnia, Tash appeared to the people of the country of Calormene and they thought that he was a god and they worshipped him. This proved their downfall.