Many rewards and blessings can be gained throughout the Islamic year, not only in Ramadan.
The blessed month of Muharram (lit. sanctified) is the first month of the Hijri calendar and is one of the four sacred months concerning which Allah Almighty Says what means: {Indeed, the number of months with Allah is twelve [lunar] months in the register of Allah [from] the day He created the heavens and the earth; of these, four are sacred. That is the correct religion, so do not wrong yourselves during them...} [Quran 9:36] Ibn Abbaas, may Allah be pleased with him, said that thisphrase (so do not wrong yourselves during them… ) referred to all the months, then these four were singled out and made sacred, so that sin in these months is more serious and good deeds bring a greater reward.

The virtue of observing supererogatory fasts during Muharram

The Messenger of Allah, salallaahu alayhi wa sallam, said, "The best of fasting after Ramadan is fasting Allah's month of Muharram." [Muslim]

The phrase "Allah's month", connecting the name of the month to the name of Allah in a genitive grammatical structure signifies the importance of the month. Al-Qaari said, “The apparent meaning is all of the month of Muharram.” But it is known that the Prophet never fasted any whole month apart from Ramadan, so this Hadeeth is probably meant to encourage increasing one's fasting during Muharram, without implying that one should fast for the entire month.
It was reported that the Prophet used to fast more in Sha'baan. It is likely that the virtue of Muharram was not revealed to him until the end of his life. [An-Nawawi and Muslim]

Al-Izz ibn Abdus-Salaam said that Allah makes certain times and placed sacred or blessed in order to encourage his servants for righteous deeds and bestow a greater reward for them. [Qawaa'id Al-Ahkaam]

'Aashooraa' in history

The tenth day of Muharram, known as 'Aashooraa', has a special significance. Ibn Abbaas, may Allah be pleased with him, said, "The Prophet, salallaahu alayhi wa sallam, came to Madeenah and saw the Jews fasting on the day of 'Aashooraa'. He said `What is this? They said, `This is a righteous day, it is the day when Allah saved the Children of Israel from their enemies, so Moosa fast-ed on this day.' He said, `We have more right to Moosa than you,' so he fasted on that day and commanded (the Muslims) to fast on that day." [Al-Bukhari]

The practice of fasting on 'Aashooraa' was known even in the days of Jaahiliyyah, before the Prophet's Message. It was reported that ‘Aa’ishah, may Allah be pleased with her, said, “The people of Jahiliyyah used to fast on that day.”

Qurtubi said, “Perhaps the Quraysh used to fast on that day on the basis of some previous law, like that of Ibraheem, may Allah exalt his mention.”

The virtues of fasting 'Aashooraa'

Ibn Abbaas, may Allah be pleased with him, said, “I never saw the Messenger of Allah so keen to fast any day and give it priority over any other time than the day of 'Aashooraa' and the month of Ramadan.” [Al-Bukhari]

The Prophet said, “For fasting the day of 'Aashoora'a, I hope that Allah will accept it as expiation for the past year.” [Muslim]

It is recommended to fast the ninth in addition to the tenth of Muharram.

Abdullaah ibn Abbaas, may Allah be pleased with her, said, "When the Messenger of Allah fasted on 'Aashooraa' and commanded the Muslims to fast as well, they said, `O Messenger of Allah, it is a day that is venerated by the Jews and Christians.' The Messenger of Allah said : `If I live to see the next year, ‘Insha'Allah’, we will fast on the ninth day too.' But it so happened that the Messenger of Allah passed away before the next year came.” [Muslim]

If a person does not know when Muharram began, and he wants to be sure of fasting on the tenth, he should assume that Thul-Hijjah was thirty days—as is the usual rule—and should fast on the ninth and tenth.

Fasting 'Aashooraa' expiates sins

Imaam an-Nawawi said that it expiates all minor sins and that "Fasting the day of 'Arafah expiates sins for two years, and the day of 'Aashooraa' expiates for one year... If there are minor sins for which expiation is desired, expiation for them will be accepted; if there are no minor sins or major sins, good deeds will be added to the servant's account and he will be raised in status... If he had committed major sins but no minor sins, we hope that his major sins will be reduced." [Al-Majmoo'ah al-Muhaththab]

Fasting 'Aashooraa' when one till has days to make up from Ramadan

The jurists differed concerning the ruling on observing voluntary fasts before making up the missed days from Ramadan or other obligatory fasts like for the fulfillment of a vow. The Hanafi jurists said that it is permissible to observe voluntary fasts before making up for the obligatory fasts, while the Maaliki and Shaafi'ee scholars say that it is permissible but Makrooh (disliked). The Hanbali scholars say that it is Haraam (forbidden) and a person must give priority to the obligatory fasts until he has made them up. [Al-Mawsoo'ah al-Fiqhiyyah]

Muslims must hasten to make up any missed fasts after Ramadan, so that they will be able to fast 'Arafah and 'Aashooraa' without any problem. If a person fasts 'Arafah and 'Aashooraa' with the intention from the night before of making up for a missed fast, this will be good enough to make up what he has missed, for the bounty of Allah is great.

Avoiding innovations on 'Aashooraa'

Shaykh al-Islam Ibn Taymiyyah was asked about the authenticity of things that people do on 'Aashooraa' to seek reward, such as wearing kohl, taking a bath, wearing henna, shaking hands with one another, cooking grains (huboob) and so on. Some other people mark this day by grieving and mourning, not drinking, wailing and rending their garments.

His reply was: "Nothing to that effect has been reported in any sound Hadeeth from the Prophet or his Companions. None of the Imaams encouraged such things, neither the four Imaams, nor any others. No reliable scholars have narrated anything like this, neither from the Prophet nor from his Companions, may Allah be pleased with them, nor from the Taabi'een; neither in any authentic report or in a weak report; neither in the books of Saheeh, nor in as-Sunan, nor in the Musnads. No Hadeeth of this nature was known during the best centuries, but some of the later narrators reported Hadeeths like: `Whoever puts kohl in his eyes on the day of 'Aashooraa' will not suffer from eye disease in that year, and whoever takes a bath on the day of 'Aashooraa' will not get sick in that year' and so on. They also reported a fabricated Hadeeth that is falsely attributed to the Prophet which says, `Whoever is generous to his family on the day of Aashooraa, Allah will be generous to him for the rest of the year.' Reporting all of this from the Prophet is tantamount to forging lies against him."'