Sometimes we are unable to fast all days during Ramadan due to various reasons, such as ill health, travel or other extenuating circumstances.
Allah is Most Merciful, and allows us the provision to make up the missed fast outside Ramadan.

Ramadan is the month during which the Quran was revealed, providing guidance for the people, clear teachings, and the statute book. Those of you who witness this month shall fast therein. Those who are ill or traveling may substitute the same number of other days. Allah wishes for you convenience, not hardship, that you may fulfill your obligations, and to glorify Allah for guiding you, and to express your appreciation.[2:185]

It may be tempting in some cases to just ignore the missed fast or simply feed a poor person to make up. While feeding a poor person is an excellent act of righteousness whether in Ramadan or otherwise, Allah recommends making up the missed fast by fasting on another day. In 2:184 Allah says, "fast is the best for you".

Specific days (are designated for fasting); if one is ill or traveling, an equal number of other days may be substituted. Those who can fast, but with great difficulty, may substitute feeding one poor person for each day of breaking the fast. If one volunteers (more righteous works), it is better. But fasting is the best for you, if you only knew.  [2:184]

We must remember, that we are not doing Allah a favor by observing the Ramadan fasts. Instead, the practices of our religion, including fasting during Ramadan, are a great gift for our souls.