The Chachamim instituted the recital of a supplementary prayer solely to make up for the preceding prayer that was missed. Therefore, one who missed both Shacharit and Minchah, due to circumstances beyond his control, prays a supplementary prayer after Ma’ariv for Minchah alone. If he desires, he may pray an additional voluntary prayer for the Shacharit that he missed (Shulchan Aruch 108:4-5), although nowadays we are not accustomed to praying voluntary prayers.
One who forgot to recite Musaf cannot make it up since the Musaf offerings are only brought on specific days. Similarly, one who forgot to pray Shacharit on a day that Musaf is recited, even though we learned that one must make up the prayer in the Amidah immediately following it, the Musaf prayer does not count and one must make up Shacharit after Minchah.
Someone who intentionally (b’meizid) missed one of the prayers cannot make it up. The Rishonim write that if he wants, he may recite it as a voluntary prayer (Shulchan Aruch 108:7). However, we already learned that today we are not accustomed to praying voluntary prayers, since one who does must be sure that he can concentrate throughout the whole prayer, from beginning to end (Kaf HaChaim 108:31).
Nonetheless, one who did not recite Minchah or Ma’ariv when he still had the chance, thinking that after he finished the matter occupying his attention, time for prayer would remain, but was ultimately drawn into what he was doing, while the time to pray passed, is not considered to have purposely negated his prayer and must make it up after the next one. The law regarding a person who started to eat before praying, with the thought that he would have time to pray afterwards, but eventually forgot, is similar. Despite the fact that he started to eat when it was forbidden (earlier in this book 12:6-7; 24:6; 25:9), since he did not contemptuously cancel his prayer, he makes it up after the next one (Shulchan Aruch 108:8). However, consider the case of a person engaged in a game, such as soccer, who was called to go pray lest the proper time pass. Out of his enthusiasm for the game, he said, “Just one more minute,” and continued to play. Meanwhile, the time to pray passed. In such a case, he cannot make up his prayer. Were it not for the game, he would have been more than happy to pray. Nevertheless, since he knew for certain that the time to pray was about to lapse, he is, indeed, deemed one who scornfully neglected his prayer.