The Halacha of "Tashlumin" allows a person to make up a prayer that he forgot to recite or was unable to recite due to circumstances beyond his control. One can make up a missed prayer only during the subsequent prayer service. For example, if a person did not recite Shaharit one morning because he overslept, felt sick, or was drunk, or if he simply forgot, then he recites an extra Amida prayer at Minha. He recites the first Amida with the intent that it counts for Minha, and he then recites the second Amida as the "make up" prayer for the prayer he missed that morning.
The Rishonim (Medieval Halachic scholars) debate the question of whether a person can make up two consecutive missed prayers. Meaning, if a person missed both the Shaharit prayer and the Minha prayer, does he recite three Amida prayers at Arbit to make up both prayers, or is he unable to make up the missed Shaharit once the time for Minha has passed? The Meiri (Rav Menahem Meiri, 1249-1315) maintained that one should indeed recite three Tefilot in such a case, whereas Tosefot (school of Medieval French and German Talmudists) held that one can make up only the most recent missed prayer. The Shulhan Aruch cites both views, and rules that a person in such a case should recite an extra Amida to make up the recent missed prayer, and then recite a third Amida on condition that if it is not warranted, it should be considered a voluntary prayer. Before reciting this third prayer, he should stipulate that according to the view that he cannot make up the first missed prayer, this Amida is recited as a strictly voluntary Tefila.
This Halacha applies even if somebody missed many months’ worth of prayers, such as if a person was, Heaven forbid, physically incapacitated for an extended period during which he was unable to pray. Another example is a case of somebody who was in prison and could not pray because the prison cell was unclean. When a person had been unable to pray for an extended period and is then able to pray, he should recite an extra Tefila to make up for the most recent prayer that he missed. Then, he should preferably recite additional Amida prayers to make up for his missed Tefilot, on condition that if they are not warranted, they should count as voluntary prayers. He should try to add a new request in each Amida prayer.
An interesting question arises in the case of a "Ba’al Teshuba" – somebody who became Torah observant after years of non-observance. Upon embracing a religious lifestyle, is he now required to make up the many years’ worth of missed Tefilot? Hacham Ovadia Yosef, as cited in Yalkut Yosef, rules that a "Ba’al Teshuba" is not required to make up his missed prayers. He notes that if the individual intentionally did not pray, despite being aware of the obligation to pray, then he certainly does not have to make up the missed prayers, since the concept of "Tashlumin" does not apply to intentionally skipped prayers. And if he was simply unaware of the obligation to pray, then he makes up for this mistake through the recitation of a single additional Amida prayer. Since he did not know of the concept of prayer, he is considered as though he missed a single prayer, which he makes up through the recitation of one additional Amida.
Summary: A person who forgot or was unable to recite a prayer recites an additional Amida at the next prayer service. If a person missed two consecutive prayers, then at the next prayer he recites an additional Amida, and then recites a third prayer on condition that it should count as a voluntary prayer. If a person could not pray for an extended period, such as in a case of a prolonged, serious illness, he should preferably recite additional Amida prayers to make up for the all prayers he missed, stipulating that they should count as voluntary prayers.
