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Well We are not really told after all
Later

it's in adventure of the empty house from the return of sherlock holmes.

Well, vast majority of theories aside, here is what happened:

Ignoring some oversights on Conan Doyle's part, Professor James Moriarty is--as is commonly quoted in attribution to him--the "Napoleon of Crime" and the ringleader of the British (and some say, European) underworld. He has connections far and wide, with resources that rival the leaders of some small countries; all this for the head of a criminal empire. Moriarty was brazen in his assertion of power. Along with subordinates that ranged from politicians to decorated military officers of old (such as former colonel and record-holding marksman, Sebastian Moran), he was as untouchable as he was infamous among the criminal element.

Sherlock Holmes was one of the few citizens with any knowledge of his limitless authority. Holmes had but his intellect and none the connections of Moriarty, of course. We are to presume Holmes had delved too deep into Moriarty's empire, and Moriarty had first attempted to have him displaced. When these attempts failed--three times--Moriarty visited Holmes at Baker Street. The two shared a (now famous, oft-quoted) exchange, in which Moriarty--in not as many words--conceded that Holmes was very clever in his ways, and only warned Holmes that, should he continue, things would not bode well for him. Holmes refused to stop and decided instead to take refuge with his old friend Dr. Watson, who had entered into private medical practice at the time.

As it turns out, Holmes had been working an operation of his own all along. He had been trying to entice the whole of Moriarty's men and connections into the hands of the police, but it had not been easy and Moriarty had began to suspect him (just not of his intention; he considered him a "meddler"). By the time Moriarty visits Holmes (and alone), Holmes has already set forth his plans into motion.

Long story short (it is fairly long for a short story, but is thrilling from front to back): Moriarty and company set fire to the rooms at Baker Street (but "no great harm was done"). Holmes, as a precaution, goes out onto the continent with Watson while a plan he's formed with the police unfolds. Unfortunately for them, Moriarty sees ahead of the plan (even if it is too late to save the majority of his men from the police) and escapes in pursuit of Holmes. As they make their way to Switzerland, Holmes begins to suspect someone is on their tail and starts to formulate a way to resolve the matter--regardless of the cost. After passing through to the small village of Merrigen, the two become acquainted with the keeper there and then set off to see the sights ahead; primarily... Reichenbach Falls.

On their way, the two are stopped by a messenger boy who brings along a note stating an English woman has fallen gravely ill at the inn and is fervently requesting an English doctor. Holmes thinks for a moment and comes to a conclusion about the situation. He sends Watson to see back to the woman. When Watson returns to the inn, however, he learns that no one there has heard about any ill English woman. Realizing what has happened, he hurries back to where he left Holmes and then to Reichenbach but sees no sign of him. After some searching, he finds a note from Holmes:

MY DEAR WATSON [it said]:

I write these few lines through the courtesy of Mr. Moriarty, who awaits my convenience for the final discussion of those questions which lie between us. He has been giving me a sketch of the methods by which he avoided the English police and kept himself informed of our movements. They certainly confirm the very high opinion which I had formed of his abilities. I am pleased to think that I shall be able to free society from any further effects of his presence, though I fear that it is at a cost which will give pain to my friends, and especially, my dear Watson, to you.

I have already explained to you, however, that my career had in any case reached its crisis, and that no possible conclusion to it could be more congenial to me than this. Indeed, if I may make a full confession to you, I was quite convinced that the letter from Meiringen was a hoax, and I allowed you to depart on that errand under the persuasion that some development of this sort would follow. Tell Inspector Patterson that the papers which he needs to convict the gang are in pigeonhole M., done up in a blue envelope and inscribed "Moriarty." I made every disposition of my property before leaving England and handed it to my brother Mycroft. Pray give my greetings to Mrs. Watson, and believe me to be, my dear fellow,

Very sincerely yours,
SHERLOCK HOLMES.

That and other indications lead him to believe both Holmes and Moriarty had tumbled over Reichenbach. Of course, "The Empty House" tells us otherwise for at least one of the pair. So, as far as the canonical story is concerned... Moriarty died at Reichenbach, and Holmes survived. That said: Sherlockian studies have always hinged on the theorizing around the canon and filling in the gaps. Take from that what you will.

Later on, after Holmes was revived but set years before (there's a canonical error here, in regards to Watson's knowledge of Moriarty's existence, but disregard that), Moriarty and his influence are again played out in the final novel, The Valley of Fear (which I personally hold to be the best-written of the four novels). There, Moriarty technically has something of a triumph over Holmes, but I'll leave you to read it and see. Let's just say... This solidifies the command that had been hinted at in "The Final Problem."

Sorry for the long post, but if you leave a Holmes question open, dangerous things can happen.