You need {proportion} of {quantity}{units} of {item}.
That is, a sentence in this format will always be grammatically correct. For example:
You need (one/a) quarter of (one/a) liter of water.
Note that when speaking about a single unit, it's common to replace one with a. For example:
You need a quarter of a liter of water.
It's worth mentioning that in some cultures, such as British English - it would also be appropriate to drop the "a" in "a quarter". But as this only applies to some proportions (quarter/half) I'd recommend an English Language Learner avoids these cases unless very comfortable with local idioms.
However, (in some/most cultures) when talking about halves - this is a special case where you can omit the of. This is simply due to common phrasing, there is no solid rule that makes this possible.
For halves, you can for example write:
You need half a liter of water.
Note that when using half without of, you will always use half a {unit} and never half one {unit}.
For completeness, it should be noted that in some cultures - there are the additional terms "quarter-{unit}" and "half-{unit}", which are short for "quarter of a {unit}" and "half of a {unit}". They would be used as "Put in a half-pint of beer" or "You need a quarter-pound of meat". Again, I'd recommend an ELL not to worry about these usages, as they can differ from region to region - but be aware of them in the case they are found in literature.
The other case you have asked about is:
you need half of the pound of potatoes.
In this case, the addition of the refers to half of a previously mentioned or specified quantity of potatoes. For example, if you're talking about a recipe that requires you to peel all the potatoes - the next step may talk about using "half of the pound of potatoes" to ensure you use those previously peeled, and not just any potatoes.
Commonly though, you will find this usage split into two parts:
Firstly, the entire quantity of the item is specified. "Take 1 pound of potatoes and peel them.".
Then later, the proportion for that specific step is used - without re-iterating the original total quantity. "Fry half of the peeled potatoes." (referring to half of the original kilogram of now peeled potatoes).
To give some additional clarity, here are some example sentences following the above rules:
You need half of a kilogram of potatoes.
You need half a kilogram of potatoes.
You need half of the kilogram of potatoes.
You need a half-kilogram of potatoes.
You need two thirds of a tonne of lead.
You need half a kilogram of flour.
You need a third of a year to complete this.
You need a quarter of a pound of meat.
You need 19/27ths of a gallon of soup.
Notes on non-standard parenthesis uses in this answer:
(A/B) is a non-optional term, but can be either A or B. e.g. "(he/she) rode the bike" must include either he or she.*
{variable} is a term to replace with anything from that category. e.g. "{proportion} of a pound" may be "quarter of a pound" etc.