Your question even with the edit is rather basic. You could easily get some more basic information on wikipedia.
But to make things clear and hopefully clear your confusion:
You are mixing two things:
The zakah or zakat on wealth
The zakat (or zakah) fard which is often quoted together with salat (prayer) in the qur'an. Here some examples and evidences for this duty from the qur'an:
And establish prayer and give zakah and bow with those who bow [in worship and obedience]. (2:43)
And establish prayer and give zakah, and whatever good you put forward for yourselves - you will find it with Allah. Indeed, Allah of what you do, is Seeing. (2:110)
Take, [O, Muhammad], from their wealth a charity by which you purify them and cause them increase, and invoke [ Allah 's blessings] upon them. Indeed, your invocations are reassurance for them. And Allah is Hearing and Knowing. (9:103)
And they who are observant of zakah (23:4)
and from the sunnah:
When the Prophet (ﷺ) sent Mu`adh to Yemen, he said to him, "You are going to a nation from the people of the Scripture, so let the first thing to which you will invite them, be the Tauhid of Allah. If they learn that, tell them that Allah has enjoined on them, five prayers to be offered in one day and one night. And if they pray, tell them that Allah has enjoined on them Zakat of their properties and it is to be taken from the rich among them and given to the poor. And if they agree to that, then take from them Zakat but avoid the best property of the people." (Sahih al-Bukhari)
This zakat is a yearly due for people who have enough wealth (of money, gold, silver, grains, cattle etc.) that reaches the nisaab (the minimum amount to be a due on this wealth) and has been in possession of this person for a hawl (a lunar year) -the majority of scholars make it a condition that this wealth has been kept in possession, so spending from it and reaching the same "value" again may not count for them-.
For example lets say: you've got money in your local currency that has a value of 10000 of your currency the nisaab these days is a bit tricky as at the time of the prophet () the nisaab for money or gold and silver was known and equal to the price or amount a certain amount of silver (595g) or a certain amount of gold (85g). At the time the difference between these two amounts was the same (as currency was either of gold or silver and one could exchange a fixed amount of silver for a piece/amount of gold), but nowadays the price of gold is much higher than that of silver so scholars these days again have a disagreement as to which nisaab one should rely on that of silver or that of gold? Let's say the actual nisaab (price of these amounts of either silver or gold depending on what you rely on) is 5670.91 of your currency in this case your savings are due to zakat and you should pay it if you have these savings with you a whole lunar year and you should pay 2.5% of them (One out of many evidences for this value from the sunnah is quoted here in a long hadith, see also fatwa islamqa #145600). If the money you have saved is less than 5670.91 of your currency then zakat is not a due on you.
The term of this zakat depends on the moment your savings have reached a time span of one lunar year being in your possession so if you've got your savings on the 2nd day of Ramadan you must pay zakat in the following year on this very day as it becomes a due on you.
The zakat related to the breaking of the fast of Ramadan
The zakat fard of 'Id al-Fitr also known as sadaqah of fitr or zakat al-Fitr (it is related to the breaking of fast on the first day of Shawwal) -see also on Wikipedia-:
This is a zakat which every Muslim who is able to feed himself and those he is in charge of on the day of 'Id must pay. It is better to pay it before the 'Id prayer takes place and it is usually paid from grains, dates etc. whatever people of the region consume the most or use the most in their food (grains for example because they are used in bread etc.), but the hanafis where among the first to allow paying the equal of it in money, which is nowadays adopted by most modern scholars. However some say it is better to pay according your own wastage for example if your daily food has a value of 50$ you should pay this amount even if the equal amount of that food is only around 10$ (but that seems to me a recommendation). As for the time to spend this zakat you should spend it at the end of Ramadan most scholars consider it best to be paid from the last night of Ramadan on. Note that this zakat is a due on every Muslim, so if you have children or parents you are in charge of you must pay for each of them the same amount.
Some evidences from the sunnah:
The Messenger of Allah (ﷺ) prescribed the sadaqah (alms) relating to the breaking of the fast as a purification of the fasting from empty and obscene talk and as food for the poor. If anyone pays it before the prayer (of 'Id), it will be accepted as zakat. If anyone pays it after the prayer, that will be a sadaqah like other sadaqahs (alms). (Sunan abi Dawod)
We used to give one Sa' of meal or one Sa' of barley or one Sa' of dates, or one Sa' of cottage cheese or one Sa' of Raisins (dried grapes) as Zakat-ul-Fitr. (Sahih al-Bukhari)
"We used to pay Sadaqatul Fitr when the Messenger of Allah was among us; a Sa' of food, or a Sa' of dates, or a Sa' of barley, or a Sa' of cottage cheese. We continued to do so until Mu'awiyah came from Ash-Sham and one of the things that he taught the people was when he said: I think that two Mudds of wheat from Ash-Sham are equivalent to a Sa' of this, So the people took to that." (Sunan an-Nasa-i)
So basically you are talking about two different kinds of zakat, one that is a due on every Muslim and one that is a due on wealthy enough Muslims under certain circumstances. As being wealthy doesn't necessary mean you must pay zakat on your wealth imam al-Laith ibn Sa'ad was a wealthy man, but it is said that he never paid zakat as he used to spent a lot of money and rarely had enough money at the end of a (lunar) year or even had debts in between.
While the conditions of zakat al-Fitr are of a kind that almost all Muslims may need to pay them at a given time.