Sadaqat ul-Fitr (also called Zakat ul-Fitr or Fitrana) is a special type of charity to purify the fast and help the poor celebrate the day of Eid. A narration from Ibn ‘Abbas says:
“The Messenger of Allah (ﷺ) enjoined Zakatul-Fitr as a purification for the fasting person from idle talk and obscenities, and to feed the poor.” (Ibn Majah, 1827)
The amount to give away is ~3.33kg of barley, or dry dates, or raisins. According to Hanafi school, an equivalent sum can be given in cash. Usually your local mosque will specify the amount in local currency.
In effect, the poor should get these funds for the day of Eid, so that they too can celebrate it.
The most common opinion is that it has to be paid before the Eid prayer. The same narration from Ibn ‘Abbas (may Allah be pleased with him) states:
“Whoever pays it before the (Eid) prayer, it is an accepted Zakah, and whoever pays it after the prayer, it is (ordinary) charity.” (Ibn Majah, 1827)
Another narration from Ibn 'Umar mentions that sahaba (may Allah be pleased with them all) used to discharge it even before the day of Eid:
"People used to give Sadaqat-ul-Fitr (even) a day or two before the Eid" (Sahih al-Bukhari, 1511)
One opinion in Hanafi school says that it is necessary to pay the Sadaqah al-fitr that have not been paid on time as soon as possible, and this position should be used by those who miss paying it on time for whatever reason.
According to Hanafi school, sadaqat ul-fitr is obligatory on any Muslim adult or child who has the wealth equal to nisab. The preferred opinion is to use silver for nisab, i.e. if one owns the equivalent of 612.36g of silver in any form of wealth aside from outstanding debts and one’s basic personal needs (such as house of residence, clothing, furniture, personal vehicle, etc).
The other three schools say it is obligatory for everyone who has surplus provisions for the night and day of Eid for himself and his dependents, i.e. Muslims who have enough food for the day of Eid and could share it.
It is advisable for every Muslim to pay it, unless one is in a difficult financial situation. Parents should also pay for their children.
Sadaqat ul-Fitr can be paid to any Muslim who is eligible to receive standard zakat. Practically speaking, it can be any Muslim person in need, starting from those closer to you, i.e. relatives, neighbors, and such. One doesn't have to specify that it is al-fitrah when passing it. If you are giving the money through an organization, it's best to make sure that the money reaches the intended recipients in time as specified above.
May Allah accept our fast, charity, and purify them from any shortcomings!