What are the voluntary fasts in Islam?

Voluntary (sunnah) fasts are extra fasts outside Ramadan that earn reward but are not required. The best known are the Day of Arafah, Ashura, six days of Shawwal, Mondays and Thursdays, and three days in the middle of each month. You can start small with one day and build up.

Why fast when it isn't required?

Voluntary fasting keeps the spirit of Ramadan alive all year, builds self-control and fills gaps in your obligatory worship. The Quran encourages extra good deeds even within the fasting verses.

فَمَن تَطَوَّعَ خَيْرًا فَهُوَ خَيْرٌ لَّهُۥ "And whoever volunteers good [i.e., excess] - it is better for him." Quran 2:184

The yearly fasts

Day of Arafah (9th Dhul Hijjah): expiates the sins of the previous and coming year (Sahih Muslim 1162). Not for pilgrims at Arafah. See Should I fast on the Day of Arafah?
Ashura (10th Muharram): expiates the previous year (Sahih Muslim 1162), ideally with the 9th as well. See What is Ashura and should I fast on it?
Six days of Shawwal: after Ramadan, like fasting the whole year (Sahih Muslim 1164).
The month of Muharram: the best fasting after Ramadan (Sahih Muslim 1163).
Sha'ban: the Prophet ﷺ fasted most of this month (Sahih al-Bukhari 1969).

The weekly and monthly fasts

Mondays and Thursdays: the Prophet ﷺ fasted Mondays, saying it was the day he was born (Sahih Muslim 1162), and deeds are shown to Allah on these days.
Three days each month: Abu Hurairah said his close friend ﷺ advised him to fast three days every month (Sahih al-Bukhari 1981). Many choose the 13th, 14th and 15th, called the "white days".
The fast of Dawud: fasting every other day, the most beloved fasting to Allah (Sahih al-Bukhari 1131). It is the maximum, not a starting point.

Days you should not fast

Fasting is forbidden on the two Eid days and, for most scholars, the three days after Eid al-Adha. Singling out Friday alone is disliked. See Can I fast on the day of Eid?

Simple rules

You can make the intention for a voluntary fast in the morning, as long as you haven't eaten since Fajr. You may also break a voluntary fast if needed; most scholars say no make-up is required, while the Hanafis say you should make it up. A married woman should check with her husband before a voluntary fast when he is at home (Sahih al-Bukhari 5195). Owing Ramadan days? Many scholars advise making those up first.

Key points

Arafah, Ashura and six days of Shawwal are the big yearly fasts.
Mondays, Thursdays and three days a month are easy regular habits.
Never fast on Eid; avoid singling out Friday.
Voluntary fasts can be started in the morning and broken if needed.

Tip: Start with one Monday a month. Once it feels normal, add the white days. Small and steady is exactly what the Prophet ﷺ loved.