Suppose someone was found guilty according to evidence, jailed, then more evidence finds him/her innocent way later (like the incriminating evidence was later found to be fabricated). Would this mean that the innocent person would recieve compensation or something else?
There is much variation between states, but a wrongful incarceration could be compensated. In almost half of the states there is no compensation. In Ohio for example, if you were charged and did not plead guilty (no plea bargaining!), and the conviction was vacated, dismissed, or reversed and the offense was "not committed by the individual or was not committed by any person", then you can sue the state, and the statute specifies $40,330 annually plus lost wages, costs, and attorney's fees.
There is a significant difference between finding that the conviction was not legally right, and that the accused is factually innocent, in this respect. If a conviction is overturned for lack of Mirandizing or reversible error by the trial judge, that doesn't lead to compensation in Ohio. If you can prove that it wasn't you (e.g. somebody else actually did it), then it could.
Another key point to consider is when the new evidence of innocence becomes available. As a general rule, a conviction can only be set aside if new evidence of innocence was not available to present at the original trial resulting in a conviction.
Suppose that Mr. Anderson is convicted of murder after a fair trial which Mr. Anderson defended primarily on the basis that the prosecution had failed to offer evidence sufficient to prove their fairly thin case beyond a reasonable doubt.
Suppose further, that Mr. Anderson chose not to take the stand in his own defense, because if he did that, the jury would have learned of Mr. Anderson's long criminal record of felony domestic violence, rape and armed robbery, and would have been biased against him as a result even if they had doubts about his innocence.
But, Mr. Anderson knew that Mr. Smith could serve as an alibi witness to establish that Mr. Anderson was on the other side of the state at the time the murder was committed but decline to call Mr. Smith because Mr. Smith and Mr. Anderson were raping an elderly woman at the time. This evidence would be very credible because Mr. Smith was recording video of the entire incident on his phone and the local town's homecoming parade which took place at the exact moment as the murder, was visible in the background of the video out a window, and many signs on floats displayed the year of this particular parade and the name of the town's high school. Mr. Smith was in a jail next to the courthouse at the time of the trial on public intoxication charges and would have been easily available to produce to testify with a subpoena. The video recording was stored on a phone in a safe deposit belonging to Mr. Anderson at the time of the trial that Mr. Anderson had easy access to because he was free on bail prior to the trial and the safe deposit box was in the same town as the trial.
Seven years later, after the statute of limitations has run on any possible prosecution for rape in the case, Mr. Anderson seeks to offer up the testimony of Mr. Smith and the video recording, in order to set aside the murder conviction for which Mr. Anderson is now on death row.
Since the evidence offered in order to set aside the conviction was not "newly discovered" after the conviction, it is not clearly established law that Mr. Anderson would have a legal right to have his conviction set aside, even if a judge believes that Mr. Anderson is "actually innocent" of the crime for which he is scheduled to be executed. It is not clear that it would be unconstitutional or illegal to execute Mr. Anderson in these circumstances, even though it was clearly established that he was actually innocent of the murder for which he was convicted and sentenced to die.
This particular issue of setting aside convictions for "actual innocence" is one of the major unresolved issues in habeas corpus law which mostly deals with the circumstances under which convictions can be set aside.