There's no exception is Islam for "students". However, if a person is extremely poor (eligible for Zakat given to him/her), then obviously he can't pay Zakat.
In your case, you're working and have an ongoing salary and I assume you're ineligible to be paid with Zakat money (i.e. can support yourself); so the due amount of annual Zakat is a mere 2.5% of your money (there are MUCH more details if you have money for commerce like assets, or if you own a land, and some other details you can lookup/ask if they apply to you. There are about 9 different types of Zakat.)
Simply stated: Annual Zakat of your money is 2.5% given to the Zakat-eligible people.
If an amount of money in one’s possession reaches the nisab and one year has elapsed, Zakat is obligatory and one must pay (2.5%) of the total amount; and to calculate it one should just multiply the amount of money by (2.5%) OR divide it by 40.
"Nisab": is the minimum amount of money for a person to have to be obliged to pay Zakat. i.e. the amount of money that declares one as ineligible to be given from Zakat money himself.
The Nisab (minimum amount of money) that one should fully own in order for him to be obliged to pay Zakat is: the equivalent amount of money (the price of) 85 grams of Gold or the equivalent amount of money (the price of) 595 grams of Silver (either of them is enough).
The calculation can get a bit difficult if you want to exactly calculate the due amount, so the easier way that people do, is that they take their whole amount of money and multiply the amount of money by (2.5%) OR divide it by 40 and they pay that result as Zakat. The easier method would result in fractions of higher amount than due amount sometimes, but Allah will reward you even for those fractions.
One last thing:
Prophet Muhammad PBUH said:
"No money is lost for charity"
i.e. you don't lose the amounts of money paid for Sadaqah ("charity" in English; and this includes Zakat).
Hadith is translated by me from the original Arabic:
"ما نقص مال من صدقة"
So in response to your comment underneath, this falls under the Zakat of Salaries.
There are two cases for this:
1- If you get a salary and you immediately spend it before the next salary is due. In this case, no Zakat is due for that salary since you spend it all and haven't saved any, and no one-year-time requirement has elapsed on that money.
2- if you get a salary, and then the next salary is due before the previous one has been spent, then that means you have money left over, or that you might have saved it. In this case, the next year, for example in January, when you get your first salary, you can quantify the amount of money you have left over/accumulated from last year, like just count all money you have, and give away 2.5% of that as Zakat.
It's difficult to keep track of what months fall under 1 and what falls under 2, so 2 covers it all. Just count how much you have each year and give 2.5% of that.
Note: Salary is any type of money that you get repeatedly over cycling periods of time.
Translated from source in Arabic from a well-renowned scholar.