There are three rules about making up i.e. performing Qaza of missed fasts.
1) If a person mistakenly eats or drinks something during the fast, then in such a case there is no Qaza for that fast, regardless of whether the fast is of Ramadhan or not.
2) If a fasting person deliberately eats or drinks, then both Qaza and Kaffara (expiation) is compulsory.
3) If a fasting person in the month of Ramadhan due to some deserving circumstances such as in journey or because of sickness, breaks the fast, then only Qaza will be liable on him/her but not the Kaffara.
The order of Kaffara is only for the fasts of Ramadhan and if someone beaks the Nafl (optional) fast due to some reason then in such case there is no expiation liable on that person. Only performing the Qaza is sufficient.
Allah Ta’ala said: No laden soul can bear another's load, (Surah Bani Israel: 15)
That is a person cannot keep a fast or offer a Salaat (prayer) on behalf of another. What it means is that fasting or offering of Salaat by another person on behalf of the one responsible doesn’t relieve the responsible of the obligation.
Allah Ta’ala said: (Fast) a certain number of days; and (for) him who is sick among you, or on a journey, (the same) number of other days; (Surah Baqarah: 184)
Journey here signifies that the duration of the journey shouldn’t be less than three days. In this verse Allah Ta’ala has given concession to the sick people and the travelers that if fasting leads to worsening of their sickness or may become a threat to their lives or if one faces hardship during the journey then during the period of his/her sickness or travel, one can break one’s fast prematurely and except those days when fasting is prohibited, one can perform the Qaza of the missed fast. The days when fasting is not permissible are five. That is the days of the two Eid and 11th, 12th and 13th days of the month of Dhul Hijjah.
Regarding the Qaza fasts of Ramadhan it is mentioned in Hidayah that the person willing to perform Qaza has got the choice to fast continuously or take breaks and fast. However, it is Mustahab (commendable) to perform the Qaza of Ramadhan fasts continuously so that the obligation is fulfilled at the earliest. If a person delays so much in performing the Qaza of them that the next Ramadhan arrives, then it is necessary that such a person should first perform the Fardh (obligatory) fasts of Ramadhan and should not hold any Qaza fast in that month and after the ending of the month should perform the Qaza of the previously missed fasts. It should be noted that delaying the Qaza fasts does not make Fidya (penalty) Wajib (Compulsory), just keeping the fasts is sufficient enough.